Covering shadows with a smaller volume

Covering shadows with a smaller v olume Daniel A. Klain Department of Mathematical Sciences Univ ersity of Massachusetts Lowell Lowell, MA 01854 USA Daniel Klain@uml.edu Lar g er things can hide behind smaller thing s. Abstract For n ≥ 2 a co nstruc tion is gi v en for con ve x bodies K and L in R n such that the ortho gonal proje ction K u can be translat ed inside L u for e ver y direction u , while the vo lumes of K and L satisfy V n ( K ) > V n ( L ) . A more general constr uction is then giv en for n -dimens ional con ve x bodies K and L such that the orthog onal projection K ξ can be translated inside L ξ for ev ery k -dimensiona l subsp ace ξ of R n , while the m -th in- trinsic v olumes of K and L satisfy V m ( K ) > V m ( L ) for all m > k . It is then sho wn that, for each k = 1 , . . . , n , there is a class of bodies C n , k such that, if L ∈ C n , k and if the orthogona l projectio n K ξ can be transla ted into L ξ for ev ery k -dimensiona l sub space ξ of R n , th en V n ( K ) ≤ V n ( L ). The families C n , k , call ed k -cylin der bodies of R n , form a strictly in- creasin g chain C n , 1 ⊂ C n , 2 ⊂ · · · ⊂ C n , n − 1 ⊂ C n , n , where C n , 1 is prec isely the collect ion of centrally symmetric compact con ve x sets in R n , while C n , n is the collectio n of all compact con v ex sets in R n . Members of each family C n , k are seen to play a fundamenta l role in rela ting cover ing conditio ns for projectio ns to the theor y of mix ed v olumes, and members of C n , k are shown to satisfy certain geometric inequa lities. Related open questio ns are also posed . Suppose that K and L are compact con ve x subsets of n -di mensional Euclidean space. For a giv en dim ension 1 ≤ k < n , suppose t hat ev ery k -dimensional orthogonal projection (shadow) of K can be t ranslated inside the corresponding projection of L . Does it follow that K h as s maller volume than L ? In t his article it is shown that the answer in general is no . It i s then shown that t he answer is yes if L is chosen from a suitable fa mily of con vex bodies that includes certain cylinders and other sets with a direct sum dec omposit ion. 1 2 Many in verse questions from con vex and int egral geometry t ake the fo llowing form: Given two conv ex bodies K and L , and two geometric in var iants f and g (such as volume, o r su rface area, or some m easure of section s or projections ), does f ( K ) ≤ f ( L ) im ply g ( K ) ≤ g ( L )? If not , then what addi tional conditi ons on K and L are necessary? These questio ns are motiva ted in part by the projection theorems of Rogers [12]. Rogers showed that if two compact con vex sets have translatio n congruent (or , m ore generally , homothetic) projections in e very linear subspace of some cho- sen dimension k ≥ 2, then the original sets K and L must be transl ation congruent (or ho mothetic). Rogers also prov ed analogous results for sections of s ets with hyperplanes through a base poi nt [12 ]. These results then set the s tage for more general (and often much m ore di ffi cult) q uestions, in which the rig id conditio ns of translation congruence or homothety are replaced with weaker con ditions, such as containment up to translation, inequalities of measure, etc. T wo not orious questions of this kin d are the Shephard Problem [15] (sol ved in- dependently by Petty [11] and Schneider [13]), and the B usemann-Petty Problem [2] (solved in work o f Gardner [3], Gardner , K old obsky , and Schlumprecht [6], and Zhang [17, 18 ]). Both questions add ress properties of bodi es K and L that are assumed to be centrally symmetric about the origin. The Shephard Problem asks: if t he ( n − 1 )-dimensional volumes of the or- thogonal projection s K u and L u of con vex b odies K and L satisfy the inequality V n − 1 ( K u ) ≤ V n − 1 ( L u ) for ev ery direction u , does it fol low that V n ( K ) ≤ V n ( L )? Although there are ready c ounter-e x amples for general (possibly non-symmetric) con vex bodies, the problem is more di ffi cult to address under the stated assump- tion that K and L are b oth centrally symmetric. In t his case Petty and Schneider hav e sh own that, whil e the answer in general is still no for di mensions n ≥ 3, the answer is yes when the con vex set L is a projection body; that is, a zonoi d. The Busemann-Petty Problem addresses th e analogous question for sections through the origin. Suppo se that con vex bodies K and L are centrall y sym metric about the o rigin. If we assum e that th e ( n − 1)-dimensional sections o f K and L satisfy V n − 1 ( K ∩ u ⊥ ) ≤ V n − 1 ( L ∩ u ⊥ ) for ev ery direction u , do es it foll ow that V n ( K ) ≤ V n ( L )? Surprisingl y the answer is no for bo dies of dimensio n n ≥ 5 and yes for b odies of di mension n ≤ 4 (see [3, 6, 17, 18]). M oreover , Lutwak [9] has sh own that, in analogy to t he Petty- Schneider theorem, t he ans wer is always yes when t he set L i s an intersection body , a construct highl y analog ous t o projection bodies (zonoids), but for which projection (th e cosine transform) is replaced in the cons truction with intersec- tion (the Radon transform). A more complete discuss ion of background t o the 3 Busemann-Petty Problem, i ts soluti on, and its variations (some of which remain open), can be found in the comprehensiv e book by Gardner [5]. Both of t he p re vious problems ass ume that bodies in q uestion are either cen- trally sy mmetric or symmetric about the o rigin; that is, K = − K and L = − L (up to t ranslation). If this elementary assu mption is omitted, then both questions are easily seen to hav e negati ve answers. For th e projection problem, compare the Reuleaux triangle, and its higher dimensional analogues, with the Euclidean ball, or compare any non -centered con vex body with its Blaschke body [5]. For the intersection problem, consider a non-centered planar set having an equ ichordal point, or the dual analogue of the Blaschke body of a non-centered set (See [5, p. 117] or [9]). In the present article we consider a related, but fundamentally di ff erent, fa mily of questions. Suppose that, instead of comparing the areas of the projections of K and L , we assume t hat the projections of L can cover t ranslates o f the proj ections of K . Specifically , suppose that, for each direction u , the ortho gonal projecti on K u of K can be transl ated so that it is contain ed i nside t he correspondi ng projecti on L u (although the required t ranslation m ay vary depending on u ). Does i t follow that K can be t ranslated so that it is contained inside L ? Does it ev en follow t hat V n ( K ) ≤ V n ( L )? These questions have easil y described negativ e answers in dim ension 2, sin ce the projections are 1-dim ensional, and con vex 1-dimensional sets have v ery little structure. (Once again, consider th e Reuleaux triangle and the circle.) The inter- esting cases begin when comparing 2-dimensional projections o f 3-dimensi onal objects, and continue from there. For higher dimensi ons, a simple example illustrates once again that K m ight not fit in side L , eve n though ev ery p rojection of L can be translated to cover the corresponding projection of K . Let L denote the unit Euclidean 3-ball , and let K denote the regular t etrahedron having edge length √ 3. Jun g’ s Theorem [1, p. 84][16, p. 320] implies t hat ever y 2-projection of K is cover ed by a translate of the unit dis k. But a simple comp utation sho ws that L do es not contain a transl ate of tetrahedron K . An analogous con struction yields a si milar result for higher dimensional simp lices and Eucli dean balls. One might say that, althou gh K can “hide behind” L from e very observer’ s p erspectiv e, this does not imply that K can hide inside L . In the p re vious counterexample it is sti ll the case that th e set L having larger (cove ring) shadows also h as l ar ger volume than K . Although the question of comparing volumes is more subt le, there are coun terexamples to this property as well. 4 This article presents the following resul ts for ev ery dimension n ≥ 2: 1 . There exist n -dim ensional con vex bodi es K and L such that t he orthogo- nal projectio n K u can be translated insi de L u for ev ery direction u , w hile V n ( K ) > V n ( L ) . 2 . There is a l ar ge class of bodies C n , n − 1 such that, if L ∈ C n , n − 1 and if K u can be translated inside L u for e very direction u , then V n ( K ) ≤ V n ( L ). In parti cular , it wi ll be shown that if the body L ha ving covering shadows is a cylinder , th en V n ( K ) ≤ V n ( L ). The m ore general collection C n , n − 1 , called ( n − 1)- cylinder bo dies, play a role for the co vering p rojection problem in analogy to that of intersection bodies for the Busemann-Petty Problem and that o f zon oids for the Shephard Problem. These results generalize to q uestions about shadows (projections) o f arbit rary lower dimension. If ξ is a k -dim ensional subs pace of R n , denote b y K ξ the orthog- onal projection of a body K into ξ . For con vex bodi es K in R n and 0 ≤ m ≤ n , denote by V m ( K ) the m th i ntrinsic volume o f K . The main theorems of this arti- cle als o yield the following m ore general observations, for each n ≥ 2 and each 1 ≤ k ≤ n − 1: 1 ′ . There exist n -dimensional con vex bodies K and L s uch that the orth ogonal projection K ξ can be translated insi de L ξ for e very k -dimensional subspace ξ o f R n , while V m ( K ) > V m ( L ) for all m > k . 2 ′ . There is a class of bodies C n , k such that, if L ∈ C n , k and if th e orthogonal projection K ξ can be translated insi de L ξ for e very k -dimensional subspace ξ o f R n , then V n ( K ) ≤ V n ( L ). The aforementioned counterexamples are constructed in Sections 2 and 3. Cylinder bodies and t heir relation t o projection and covering are described in Sections 4 and 5, leading to the Shadow Containment Theorem 5 .3, which relates cove ring of s hadows to a family of inequaliti es for mixed volumes. These devel- opments lead in turn to Theorem 6.1, where it i s shown th at, if ev ery shadow of a cylinder bo dy L contains a translate of the correspondi ng s hadow of K , then L must hav e greater v o lume than K . In Section 7 the counterexample constructions of Sections 2 and 3 are used t o prove a family of g eometric in equalities satis fied by memb ers of each collection C n , k . Section 8 uses Theorem 6.1 t o prove that V n ( K ) ≤ nV n ( L ) whene ver the proj ections o f K can be t ranslated ins ide those o f L . The constructi ons and theorems of this article mo tiv ate a num ber of new open questions related to covering projections, some of which are p osed in the final section. 5 1. P relimin ar y ba ckgr ound Denote by K n the set of compact con vex subsets of R n . The n -dimensional (Euclidean) volume of a con vex set K will be denoted V n ( K ). If u is a uni t vector in R n , denote by K u the orthogonal projection of a set K o nto the subspace u ⊥ . Let h K : R n → R denote the support functi on o f a com pact con vex set K ; that is, h K ( v ) = max x ∈ K x · v If u is a un it vector in R n , denote by K u the support set o f K in the direction of u ; that is, K u = { x ∈ K | x · u = h K ( u ) } . If P is a con ve x poly tope, then P u is the face of P having u i n its outer normal cone. Giv en two compact con vex sets K , L ∈ K n and a , b ≥ 0 denote a K + b L = { a x + by | x ∈ K and y ∈ L } An e xp ression of this form is called a Minkow ski combination or Minkowski sum . Because K and L are con vex, the set a K + b L is also con vex. Con vexity also implies that a K + b K = ( a + b ) K for all a , b ≥ 0. Support functions are easily seen to satisfy th e identity h aK + bL = a h K + bh L . Moreover , the volume of a Mi nko wski com bination of two compact con vex sets is giv en by Steiner’ s formu la: (1) V n ( aK + b L ) = n X i = 0 n i ! a n − i b i V n − i , i ( K , L ) , where the mixed volu mes V i , n − i ( K , L ) depend only on K and L and the indices i and n . In particular , if we fix t wo con vex s ets K and L then the functi on f ( a , b ) = V n ( aK + bL ) is a hom ogeneous polynom ial of degree n in t he non- negati ve variables a , b . Each mixed v olume V n − i , i ( K , L ) is non-negative, continuo us in the entries K and L , and monot onic wit h respect to set inclusio n. Note also that V n − i , i ( K , K ) = V n ( K ). If ψ is an a ffi ne transformation whose linear compo nent has determinant denoted det ψ , then V i , n − i ( ψ K , ψ L ) = | det ψ | V n − i , i ( K , L ). If P is a polytope, th en the mixed v olum e V n − 1 , 1 ( P , K ) s atisfies the classical “base-height” formula (2) V n − 1 , 1 ( P , K ) = 1 n X u ⊥ ∂ P h K ( u ) V n − 1 ( P u ) , where this s um is finite, tak en over all outer normals u to the facets on the bound- ary ∂ P . These and many o ther properties of con ve x bodies and m ixed volumes are described in detail in each of [1, 14, 16]. 6 The Brunn-Minkowski inequality asserts that, for 0 ≤ λ ≤ 1, (3) V n ((1 − λ ) K + λ L ) 1 / n ≥ (1 − λ ) V n ( K ) 1 / n + λ V n ( L ) 1 / n . If K and L hav e int erior , th en equali ty hol ds in (3) if and only if K and L are homothetic; that is, i ff there exist a ∈ R and x ∈ R n such that L = a K + x . On combining (3) with Steiner’ s formula (1) one obtains the Minkow ski mixed volume inequality: (4) V n − 1 , 1 ( K , L ) n ≥ V n ( K ) n − 1 V n ( L ) , with the same equality conditions as i n (3). See, for example, any of [1, 4 , 14, 16]. If K ∈ K n has non-empty i nterior , define the surf ace are a measur e S K on the ( n − 1)-dimension al unit sphere S n − 1 as fol lows. For A ⊆ S n − 1 denote by K A = S u ∈ A K u , and define S K ( A ) = H n − 1 ( K A ), the ( n − 1)-dim ensional Hausdor ff measure of the subset K A of the boundary of K . (See [14, p. 20 3].) Note that, if P i s a pol ytope, then S P is a po inted measure concentrated at precisely those directions u that are outer normals to the facets of P . The measure S K is easily shown to satisfy the property that (5) Z S n − 1 u d S K = ~ o , that is , the m ass distribution on th e sphere d escribed by S K has center o f m ass at the origin. The identity (2) can now be e x pressed in its more general form: (6) V n − 1 , 1 ( K , L ) = 1 n Z S n − 1 h L ( u ) d S K ( u ) , for all conv ex bodies K and L such that K has non-empty interior . It follows from (6) and the Minkowski linearity of the s upport function t hat, for K , L , M ∈ K n and a , b ≥ 0, (7) V n − 1 , 1 ( K , aL + b M ) = aV n − 1 , 1 ( K , L ) + bV n − 1 , 1 ( K , M ) . If B is a unit Euclidean ball centered at th e origin, then h B = 1 in ev ery direction, so that nV n − 1 , 1 ( K , B ) = S ( K ), th e surface ar ea of the con vex body K . Minkowski’ s Existence Theorem [1, p. 125][14, p. 390] gives an important and useful con verse to the i dentity (5): If µ is a non-negative measure on t he unit sphere S n − 1 such that µ has center of mass at the origin, and if µ is not concentrated on any great (equatorial) ( n − 1)-subsphere, then µ = S K for so me K ∈ K n . Moreover , this con ve x body K is uniq ue up to translation . Minkowski’ s Existence Theorem provides the frame work for the fol lowing def- inition: For K , L ∈ K n and a , b ≥ 0 , defi ne the Blaschk e combination a · K # b · L to be the unique con vex body (up to translation) such that S a · K # b · L = aS K + bS L . 7 Although t he Blaschke sum K # L is identical (up to translation) to the Minkowski sum K + L for conv ex bodies K and L in R 2 , the two sums are sub stantially di ff erent for bodies in R n where n ≥ 3. Moreover , for dimens ion n ≥ 3, the scalar multipli cation a · K also di ff ers from the usual scalar multipli cation aK used wi th Minkowski combinations . Specifically , a · K = a 1 n − 1 K , si nce surface area in R n is homogeneous of degree n − 1. It follows from (6) that, for K , L , M ∈ K n and a , b ≥ 0 , (8) V n − 1 , 1 ( a · K # b · L , M ) = a V n − 1 , 1 ( K , M ) + bV n − 1 , 1 ( L , M ) . Note the important di ff erence between (7) and (8) for n ≥ 3. It i s not di ffi cult to show that every polytope is a Blaschke combinat ion of a finite number of simplices, while e very c entrally symmetric polytope is B laschke combination of a finite num ber of parallelot opes (i.e., a ffi ne images o f cubes) [7, p. 334]. A st andard continuity argument (usi ng the Minkowski Existence Theo- rem and t he selection principle for con vex bodies [14, p. 50]) then implies t hat e very c on vex bod y can be approximated (in the Hausdor ff topology) by Blaschk e combinations of simplices, wh ile ev ery centrally s ymmetric con vex body can b e approximated by Blaschke combinations of parallelotopes. A brief and elegant d iscussion of Blaschke sums, their properties, and applica- tions, can be also be found in [9]. 2. A counterexample W e w ill exhibit con vex bodies K and L in R n , such that V n ( K ) > V n ( L ), while the orthogonal p rojection L u contains a translate of t he corresponding projection K u for each unit direction u . Note that a suitable disk and Reuleaux triangle provide a well-known coun- terexample in the 2-dim ensional case. This sectio n provides examples for bodies of dimension n ≥ 3. For K ∈ K n , denote by r K the inradius of K ; that is, the maximum radius taken over all Euclidean balls i nside K . Denote by d K the minimal width of K ; that is, the minimum length taken over all orthogonal projections o f K onto lines through the origin. The minimal width is also equal to the minimum distance b etween any two parallel support planes for K . Let ∆ denote the n -di mensional regular si mplex h a ving unit edg e length. The following well-known statistics will be used in the construction that follows: 8 τ n = volume of ∆ = √ n + 1 2 n / 2 n ! S ( ∆ ) = surface area of ∆ = ( n + 1) √ n 2 n − 1 2 ( n − 1)! = ( n + 1) τ n − 1 r ∆ = in radius of ∆ = 1 √ 2 n ( n + 1) = n τ n S ( ∆ ) and (9) d ∆ = minim al width of ∆ =            2( n + 1) √ n + 2 r ∆ 2 √ n r ∆ =                q 2( n + 1) n ( n + 2) if n is e ven q 2 n + 1 if n is odd See, for example, [1, p. 86]. T o construct and verify the counterexample it wil l be necessary t o compare the minimal width and in radius o f a re gular simplex wit h th ose of i ts l ower - dimensional projections. Steinh agen’ s inequalit y asserts that, f or K ∈ K n , (10) r K ≥          √ n + 2 2 n + 2 d K if n is e ven 1 2 √ n d K if n is odd A p roof of (10) is gi ven in [1, p. 86]. If u is a uni t vector , then the orthogonal projection ∆ u satisfies d ∆ u ≥ d ∆ , where d ∆ u is now computed from withi n the ( n − 1)-dimensi onal subspace u ⊥ . Since d im( ∆ u ) = n − 1 has parity opp osite that of n , it follows from (10) that r ∆ u ≥          √ n + 1 2 n d ∆ u 1 2 √ n − 1 d ∆ u ≥          √ n + 1 2 n d ∆ if n is odd 1 2 √ n − 1 d ∆ if n is e ven Combining this with (9) yields (11) r ∆ u ≥            1 n √ 2 if n is odd √ n + 1 √ 2 √ n ( n − 1)( n + 2) if n is e ven            ≥ 1 n √ 2 Let B n denote the n -dimensional Euclidean ball centered at the origin and ha v- ing unit radius. For 0 ≤ ǫ ≤ 1, denote K ǫ = ǫ ∆ +  1 − ǫ n √ 2  B n . 9 Pr oposition 2.1. F or e ach unit vec tor u in R n , ther e exists v ∈ u ⊥ such that K ǫ u + v ⊆ ∆ u In o ther words, each sh adow of the sim plex ∆ contains a translate of the corre- sponding shadow of K ǫ . Pr oof. Let u be a unit vector in R n . Since 1 n √ 2 ≤ r ∆ u , there e xists w ∈ u ⊥ such that 1 n √ 2 B n − 1 ⊆ ∆ u − w . Hence, K ǫ u = ǫ ∆ u + (1 − ǫ ) 1 n √ 2 B n − 1 ⊆ ǫ ∆ u + (1 − ǫ )( ∆ u − w ) = ∆ u + ( ǫ − 1) w . Setting v = (1 − ǫ ) w , we hav e K ǫ u + v ⊆ ∆ u .  Next, rec all from Steiner’ s formula (1) that if K is a con ve x body in R n then (12) V n ( ǫ K + α B n ) = ǫ n V n ( K ) + ǫ n − 1 α S ( K ) + α 2 f ( α, ǫ ) , where f ( α, ǫ ) is a polynomi al in α and ǫ having non-negati ve coe ffi cients. Pr oposition 2.2. If 1 − ǫ > 0 is su ffi cientl y small, then V n ( K ǫ ) > V n ( ∆ ) . Pr oof. W e n eed to show that V n ( K ǫ ) − V n ( ∆ ) > 0. Applying (12) yields V n ( K ǫ ) − V n ( ∆ ) = V n  ǫ ∆ + 1 − ǫ n √ 2 B n  − V n ( ∆ ) = ( ǫ n − 1 ) V n ( ∆ ) + ǫ n − 1  1 − ǫ n √ 2  S ( ∆ ) +  1 − ǫ n √ 2  2 f n ( ǫ ) = ( ǫ n − 1 )  √ n + 1 2 n / 2 n !  + ǫ n − 1  1 − ǫ n √ 2   ( n + 1) √ n 2 n − 1 2 ( n − 1)!  +  1 − ǫ n √ 2  2 f n ( ǫ ) = ( ǫ n − 1 )  √ n + 1 2 n / 2 n !  + ǫ n − 1 (1 − ǫ )  ( n + 1) √ n 2 n / 2 n !  +  1 − ǫ n √ 2  2 f n ( ǫ ) where f n ( ǫ ) is a pol ynomial in ǫ . It follows that V n ( K ǫ ) − V n ( ∆ ) > 0 i f and only if ǫ n − 1 (1 − ǫ )  ( n + 1) √ n 2 n / 2 n !  +  (1 − ǫ ) 2 2 n 2  f n ( ǫ ) > (1 − ǫ n )  √ n + 1 2 n / 2 n !  if and only if (13) ǫ n − 1 p n ( n + 1) + 2 n / 2 n !  1 − ǫ 2 n 2 √ n + 1  f n ( ǫ ) > (1 + ǫ + ǫ 2 + · · · + ǫ n − 1 ) As ǫ → 1, the left-hand si de o f (13 ) approaches √ n ( n + 1), whi le the right-hand side approaches n , a strictly smaller v alue for all po sitive integers n . It follows that V n ( K ǫ ) > V n ( ∆ ) for ǫ su ffi ciently close to 1.  10 Propositions 2. 1 and 2.2 imply that if 0 < ǫ < 1 i s su ffi ciently close to 1, then e very shadow of ∆ cont ains a translate of the corresponding shadow of the bod y K ǫ , ev en though V n ( K ǫ ) > V n ( ∆ ). More precise conditi ons on adm issible values o f ǫ depend on n . For the case n = 3 the inequalit ies u sed in the pro of o f Propositio n 2.2, along with some ad- ditional very crude estimates, imply that ǫ = 0 . 9 gi ves a specific counterexample. In other words, the 3-dimensional con ve x bodi es: K = 9 10 ∆ 3 + 1 30 √ 2 B 3 and ∆ 3 hav e th e property that each shadow of K can be cove red by a translate of the cor- responding shadow of the unit re gu lar tetrahedron ∆ 3 , eve n though K has gr eater volume 1 than ∆ 3 . At this point one mi ght ask whether suitable condition s on either of the bodi es K and L might g uarantee th at cover ing shadows implies larger volume. It is not di ffi cult to show that if L is centrally sym metric, then L will have greater v o lume than K when the shadows of L can cov er those o f K . T o see this, suppose that L = − L . If K u ⊆ L u + v , t hen − K u ⊆ − L u − v = L u − v so th at K u + ( − K u ) ⊆ L u + v + L u − v = L u for every d irection u . It foll ows th at K + ( − K ) ⊆ L + L = 2 L . Monoton icity of volume and the Brunn-Minkowski Inequality (3) then imply that V n ( L ) 1 / n ≥ V n  1 2 K + 1 2 ( − K )  1 / n ≥ 1 2 V n ( K ) 1 / n + 1 2 V n ( − K ) 1 / n = V n ( K ) 1 / n , so that V n ( L ) ≥ V n ( K ). This volume inequality also turns out hold when L is chosen from a much lar ger family of bodies, to be described in Sec tion 6. 3. A more general counterexample The counterexample of Section 2 will now be generalized. If ξ is a k - dimensional subspace of R n , denote by K ξ the orthogonal projection of a s et K ⊆ R n to the subs pace ξ . For 0 ≤ m ≤ n denote by V m ( K ) the m th intrinsic volume of K . The intrinsic volume funct ional V m restricts t o m -dim ensional vol- ume o n m -dim ensional con vex sets and is proportion al to t he m ean m -volume of the m -dimens ional orthogonal p rojections o f K for more general K ∈ K n . See, for example, [8] or [14, p. 210]. The following lemma is helpful for e xt ending some lo w dimensional construc- tions to higher dimensio n. 1 A more precise calculation yields V 3 ( K ) ≈ 0 . 122 and V 3 ( ∆ 3 ) ≈ 0 . 1 18. 11 Lemma 3.1. Suppose that K and L a r e compact conv e x sets in R j ⊆ R n , wher e j ≤ n, and s uppose that L ξ can cover K ξ for all i -subspaces ξ of R j . Then L ξ can cover K ξ for all i-subspaces ξ of R n . In other words, i f the i -dim ensional shadows of L can cover th ose of K in R j , then this covering relation is preserved when K and L are embedded together (along with R j ) in the higher dimensional space R n . Pr oof. Suppose that K and L are compact conv ex sets in R j , and suppose that L ξ can cove r K ξ for all i -subspaces ξ of R j . Suppose that η i s an i -dimensional subspace of R j + 1 . Then dim( η ⊥ ) = j − i + 1, and dim( η ⊥ ∩ R j ) = j − i for generic choices of η . Assu me η is chosen thi s way . Let ξ denote the orthogonal complement of η ⊥ ∩ R j taken within R j . Since dim( ξ ) = i , there exists v ∈ R j such th at ( K + v ) ξ ⊆ L ξ , by t he covering assumption for K and L in R j . This means that, for each x ∈ K + v , there exists y ∈ L su ch that x − y is orthogonal to ξ . It follows from the constructio n of ξ that x − y ∈ η ⊥ . Hence, for all x ∈ K + v , th ere exists y ∈ L such that x − y ∈ η ⊥ . T his implies that K η + v η ⊆ L η . W e have shown that L η can cov er K η for all i -subspaces η of R j + 1 such that dim( η ⊥ ∩ R j ) = j − i . Sin ce this is a dense family of i -subspaces, the lemma follows more g enerally for all i -subspaces o f R j + 1 . By a suitable iteration of t his ar gument, the lemma then follows for i -subspaces of R n , for any n > j .  W e can no w generalize the counterexample of S ection 2. Theor em 3.2. Let n ≥ 3 and 1 ≤ k < n. Ther e e xist con vex bodies K , L ∈ K n such that L ξ can cover K ξ for all k -dimensional subsp aces ξ , while V m ( K ) > V m ( L ) for all m > k. Note th at this theorem is a lready well-known for the case k = 1 . In that par ticu- lar case, the covering condition merely asserts that the width of K i n any direction is smaller than or equal to the correspondin g width of L . The novel aspect of this result addresses the cases in which 2 ≤ k < n . Pr oof. If k = 1, t hen let K be an n -si mplex, and let M = 1 2 K + ( − 1 2 K ), the di ff er- ence body of K . It then follows from the Minkowski mixed volume inequality (4) and the classical mean projection formulas for intrinsi c v olumes [5, 8, 14] that V m ( M ) > V m ( K ) for m ≥ 2, while K and M ha ve identical wi dth in e very direc- tion. Now suppose that k ≥ 2. Let ˆ K and ˆ L be chosen i n K k + 1 so that ˆ L ξ can cover ˆ K ξ for all k -subsp aces ξ of R k + 1 , whi le V k + 1 ( ˆ K ) > V k + 1 ( ˆ L ). (One could follow the explicit construction gi ven in Sec tion 2, for example.) 12 If n = k + 1, we are done. If n > k + 1, embed ˆ K and ˆ L in R n via the usual coordinate em bedding of R k + 1 in R n . Then ˆ L ξ can cov er ˆ K ξ for all k -subsp aces ξ of R n , by Lemma 3.1. Let C denote the unit cube in R n − k − 1 with edges parallel to the standard axe s in the orthogonal complement to R k + 1 in R n . Let K = ˆ K + ǫ C and L = ˆ L + ǫ C . Then L ξ can cove r K ξ for all k -dimensional subspaces ξ once again, si nce K ξ = ˆ K ξ + ǫ C ξ , and similarly for L . Moreover , if m ≥ k + 1 then V m ( K ) = V m ( ˆ K + ǫ C ) = X i + j = m V i ( ˆ K ) V j ( C ) ǫ j by the Cartesian product formula for intrinsic volumes [ 8, p. 130]. Hence, V m ( K ) = k + 1 X i = 0 V i ( ˆ K ) V m − i ( C ) ǫ m − i = k + 1 X i = 0 n − k − 1 m − i ! V i ( ˆ K ) ǫ m − i = ǫ m − k − 1 n − k − 1 m − k − 1 ! V k + 1 ( ˆ K ) + f K ( ǫ ) where f K ( ǫ ) i s a polynomi al in ǫ composed of m onomials ha ving degree greater than m − k − 1. A simi lar formula holds for V m ( L ). Therefore, V m ( K ) − V m ( L ) = ǫ m − k − 1 n − k − 1 m − k − 1 !  V k + 1 ( ˆ K ) − V k + 1 ( ˆ L )  + ( f K ( ǫ ) − f L ( ǫ )) , where f K ( ǫ ) − f L ( ǫ ) i s a poly nomial in ǫ com posed of monom ials h a ving degree greater than m − k − 1. Since the lowest degree coe ffi cient of the pol ynomial formula for V m ( K ) − V m ( L ) is posit iv e, we have V m ( K ) − V m ( L ) > 0 when ǫ > 0 is su ffi ciently small.  4. C ylinders and shado w co vering Let K ∈ K n and suppos e th at P ∈ K n is a polyt ope. A facet of P is a face (support set) of P having dimension n − 1. W e say that P cir cumscribes K if K ⊆ P and K also m eets e very facet of P . Lemma 4.1 (Circumscribing Lemma) . Let K , P ∈ K n , wher e P is a pol ytope. If P cir cumscribes K then (14) V n − 1 , 1 ( P , K ) = V n ( P ) . If we ar e given that K ⊆ P , then (14) holds if and only if P cir cumscribes K . 13 Pr oof. If K ⊆ P and if K meets ev ery facet o f P , th en h K ( u ) = h P ( u ) whenev er the direction u is normal to a facet of P . Since P i s a polytope, the mixed v olum e formula (2) yields V n − 1 , 1 ( P , K ) = 1 n X u ⊥ ∂ P h K ( u ) V n − 1 ( P u ) = 1 n X u ⊥ ∂ P h P ( u ) V n − 1 ( P u ) = V ( P ) . Con versely , i f we are given that K ⊆ P , th en h K ( u ) ≤ h P ( u ), wi th equality for all f acet normals u if and only if P circumscribes K , s o that (14) holds if and only if P circum scribes K .  The case in whi ch P is a simpl ex is especially impo rtant, because o f the fol- lowing t heorem of Lutwak [10] (see also [8]), itself a cons equence of Helly’ s theorem. Theor em 4.2 (Lutwak’ s Containment Theorem) . Let K , L ∈ K n . Suppo se t hat, for every simple x ∆ such th at L ⊆ ∆ , ther e is a vector v ∆ ∈ R n such that K + v ∆ ⊆ ∆ . Then ther e is a vector v ∈ R n such that K + v ⊆ L. Lutwak’ s theorem combin es wit h the Circumscribing Lemma to yield the fol- lowing useful corollary (also from [10]). Corollary 4.3. Let K , L ∈ K n . The inequalit y V n − 1 , 1 ( ∆ , K ) ≤ V n − 1 , 1 ( ∆ , L ) holds for all simplices ∆ , if and only if ther e exists v ∈ R n such that K + v ⊆ L. Suppose t hat λ 1 ≥ λ 2 ≥ . . . ≥ λ m > 0 are pos itive integers such t hat λ 1 + λ 2 + . . . + λ m = n . Denote λ = ( λ 1 , λ 2 , . . . , λ m ). The vector λ is someti mes called a partition of th e posit iv e integer n . Using this notation, t he size of the largest part of any partition λ is given by th e first entry λ 1 . A con vex body K ∈ K n will be called λ -decompos able if there exists a ffi ne subspaces ξ i of R n such that dim ξ i = λ i and R n = ξ 1 ⊕ · · · ⊕ ξ m and if there exists compact con vex sets K i ⊆ ξ i such that K = K 1 + · · · + K m . In this case we wil l write K = K 1 ⊕ · · · ⊕ K m The bo dy K will b e called λ -ortho-decompos able if ξ i ⊥ ξ j for each i , j . For example, a cylinder i s an ( n − 1 , 1)-decompos able bod y . A (1 , 1 , . . . , 1)- decomposable b ody is a parallelotope, while a (1 , 1 , . . . , 1)-ortho-decomp osable body is an orthogonal box. For k ∈ { 1 , . . . , n − 1 } , d enote by G ( n , k ) the collection of all k -dimensional linear subspaces ξ of R n , sometimes called the ( n , k )-Grassmannian. For ξ ∈ G ( n , k ) and K ∈ K n , denote by K ξ the orthogonal projection of the body K onto the subspace ξ . 14 Lutwak’ s Contain ment Theorem 4.2 and its Corollary 4.3 lead to the following useful condition for determining when the shadows of one body can cover t hose of another . Theor em 4.4 ( First Shado w C ontainment Theorem) . Let K , L ∈ K n , and suppos e that 1 ≤ k ≤ n − 1 . The orthogonal pr ojections L ξ of L can cover the corr esponding pr ojection s K ξ of K f or all ξ ∈ G ( n , k ) if and only if V n − 1 , 1 ( C , K ) ≤ V n − 1 , 1 ( C , L ) for all λ -ortho-decomposable C ∈ K n such that λ 1 ≤ k. Pr oof. Suppose that C i s a λ -ortho-decomposable polytop e, with ort hogonal de- compositio n C = a 1 C 1 ⊕ · · · ⊕ a m C m , wh ere each C i has a ffi ne hull parallel to a subspace ξ i of dimension λ i and a 1 , . . . , a m > 0. Note that V n ( C ) = V λ 1 ( a 1 C 1 ) · · · V λ m ( a m C m ) = a λ 1 1 · · · a λ m m V λ 1 ( C 1 ) · · · V λ m ( C m ) , since the decomposition is orthogonal. It follows from (2) that V n − 1 , 1 ( C , K ) = 1 n X u ⊥ ∂ C h K ( u ) V n − 1 ( C u ) where th e sum i s taken over all unit di rections u ∈ R n normal to facets o f C . The product structure of C t hen implies that V n − 1 , 1 ( C , K ) = 1 n m X i = 1 X u ⊥ ∂ C i h K ( u ) V λ 1 ( a 1 C 1 ) · · · V λ i − 1 ( a i C u i ) · · · V λ m ( a m C m ) where, for each i , t he inner s um i s t aken over all unit directions u ∈ ξ i normal to facets of C i . Hence, V n − 1 , 1 ( C , K ) = 1 n m X i = 1 V n ( C ) V λ i ( a i C i ) X u ⊥ ∂ C i h K ( u ) V λ 1 − 1 ( C u i ) a λ i − 1 i = 1 n m X i = 1 iV n ( C ) a i V λ i ( C i ) V λ i − 1 , 1 ( C i , K ξ i ) for all a 1 , . . . , a m > 0. If L ξ can cover K ξ for all ξ ∈ G ( n , k ), then L η also can cov er K η for all lower dimensional subspaces η ∈ G ( n , j ), where 1 ≤ j ≤ k . In parti cular L ξ i can cover K ξ i for each i , since dim ξ i = λ i ≤ λ 1 = k . It follows that each V i − 1 , 1 ( C i , K ξ i ) ≤ V i − 1 , 1 ( C i , L ξ i ) by t he m onotonicit y and t ranslation in variance of mixed volumes. Therefore, V n − 1 , 1 ( C , K ) ≤ V n − 1 , 1 ( C , L ) for all λ -ortho-decomposable p olytopes C . 15 The inequ ality then holds for arbitrary λ -ortho-decomposable C by continuit y of mixed v olumes. Con versely , if V n − 1 , 1 ( C , K ) ≤ V n − 1 , 1 ( C , L ) for all λ -ortho-decomp osable C ∈ K n such that λ 1 ≤ k , then 1 n m X i = 1 iV n ( C ) a i V λ i ( C i ) V λ i − 1 , 1 ( C i , K ξ i ) ≤ 1 n m X i = 1 iV n ( C ) a i V λ i ( C i ) V λ i − 1 , 1 ( C i , L ξ i ) for all such C and all a 1 , . . . , a m > 0. In particular , V λ i − 1 , 1 ( ∆ , K ξ i ) ≤ V λ i − 1 , 1 ( ∆ , L ξ i ) for every λ i -simplex ∆ in e very λ i -dimensional subspace ξ i of R n so that L ξ i can cove r K ξ i by Corollary 4.3.  5. C ylinder bodies and shadow co v ering So far we have restricted attention to o rthogonal cylinders and decompos- able sets. Howe ver , the previous resul ts generalize easily to arbitrary (possibly oblique) cylinders and dec ompositi ons. For S ⊆ R n and a nonzero vector u , let L S ( u ) denote the set of straight l ines in R n parallel to u and meeting the set S . Pr oposition 5.1. Let K , L ∈ K n . Let ψ : R n → R n be a non-sin gular l inear transformation . Then L u contains a translate of K u for all unit dir ections u if and only if ( ψ L ) u contains a translate of ( ψ K ) u for all u. Pr oof. The projection L u contains a transl ate o f K u for each unit vector u if and only if, for each u , there exists v u such that (15) L K + v u ( u ) ⊆ L L ( u ) . But L K + v u ( u ) = L K ( u ) + v u and ψ L K ( u ) = L ψ K ( ψ u ). It follows that (15) holds i f and only if L K ( u ) + v u ⊆ L L ( u ), if and only if L ψ K ( ψ u ) + ψ v u ⊆ L ψ L ( ψ u ) for all uni t u , Set ˜ u = ψ u | ψ u | and ˜ v = ψ v u . The relation (15) now holds if and only if, for all ˜ u , there exists ˜ v such that L ψ K ( ˜ u ) + ˜ v ⊆ L ψ L ( ˜ u ) , which holds if and only if ( ψ L ) ˜ u contains a translate of ( ψ K ) ˜ u for all ˜ u .  W e are no w in a position to define a much lar ger family of objects that serve to generalize the Shadow Containment Theorem 4.4. 16 Definition 5.2. F or each k ∈ { 1 , . . . , n } denote by C n , k set of all bodi es K ∈ K n that can be app r oximated (i n the usual Hausdo r ff topology) b y Blaschk e combinat ions of λ -decomposable sets for any λ such t hat λ 1 ≤ k. Elements of C n , k will be called the k -cylinder bodies of R n . Recall that any centrally symm etric polyt ope is a Blaschke sum of parallelo- topes. It follows that C n , 1 is precisely the set of all centrally symmetric con ve x bodies in R n . For n ≥ 3 and k ≥ 2, the cylinder bod ies C n , k are a lar ger fam- ily of objects. For example, a triangu lar cylinder in R 3 lies in C 3 , 2 , but not in C 3 , 1 , since it is not centrally symm etric. Note also that C n , k is clos ed u nder a ffi ne transformations. The definition of C n , k depends on the ambi ent dimension n as well as th e va lue k , because th e notion of Blaschke sum # depends on n . For example, while Minkowski sum satisfies the projection identity ( K + L ) ξ = K ξ + L ξ for subspaces ξ ⊆ R n , the analogous statement need not hold for Blaschke summation. Note also that C n , n = K n by definition. Moreover , i t follows from the d efinition that C n , i ⊆ C n , j whenev er i ≤ j . It wi ll be sh own in Section 6 t hat C n , i is a pr oper subset of C n , j when i < j . In parti cular , it will b e seen that full-dimensi onal simplices are not k -cylinder b odies of R n for a ny k < n . A necessary condition for being a k -cylinder body will be described in Section 7. The significance of each col lection C n , k is described in part by the following theorem. Theor em 5.3 (Second Shado w Containment Theorem) . Let K , L ∈ K n and let 1 ≤ k ≤ n. The fol lowing ar e equivalent: (i) The or thogonal projections L ξ of L can cover the corr esponding pr ojec- tions K ξ of K f or all subspaces ξ ∈ G ( n , k ) . (ii) The a ffi ne pr ojections π L of L can cover the corr esponding pr ojections π K of K for all a ffi ne pr ojections π of rank k. (iii) V n − 1 , 1 ( C , K ) ≤ V n − 1 , 1 ( C , L ) for all λ -ortho-decomposable s ets C su ch that λ 1 ≤ k. (iv) V n − 1 , 1 ( C , K ) ≤ V n − 1 , 1 ( C , L ) for all λ -decomposable s ets C such that λ 1 ≤ k. (v) V n − 1 , 1 ( Q , K ) ≤ V n − 1 , 1 ( Q , L ) f or all k-cylinder bod ies Q ∈ C n , k . Pr oof. The equiva lence o f (i) and (ii) follows from Propos ition 5.1 . The equiva- lence of (i) and (iii) follows from Theorem 4.4. T o show that (iii) impli es (iv) , suppose that (iii ) h olds for the pair K , L . It follows from (i) and Proposi tion 5.1 that (i) also holds for the pair of bodies 17 ψ − 1 K , ψ − 1 L , for any non-singular a ffi ne t ransformation ψ . Therefore (iii) also holds for the pair of bodies ψ − 1 K , ψ − 1 L ; that is, V n − 1 , 1 ( C , ψ − 1 K ) ≤ V n − 1 , 1 ( C , ψ − 1 L ) . for all λ -ortho-decomposabl e set s C such that λ 1 ≤ k . Let u s suppose that ψ has unit d eterminant. Then V n − 1 , 1 ( ψ C , K ) = V n − 1 , 1 ( C , ψ − 1 K ), and si milarly for L , by the a ffi ne in variance of (mixed) volumes, so that V n − 1 , 1 ( ψ C , K ) ≤ V n − 1 , 1 ( ψ C , L ) . for all λ -orth o-decomposable sets C such th at λ 1 ≤ k . If C ′ is a λ -decom posable set, then C ′ = ψ C for some λ -ortho-decomposable set C and some a ffi ne transfor - mation ψ of unit determinant. (iv) no w follows. (iv) implies (v) by the Blaschke-linearity of th e functional V n − 1 , 1 ( · , · ) in i ts first parameter and the continuit y of V n − 1 , 1 . Finally , (v) implies (iv) , and (iv) implies (iii) , in both cases a fortior i .  6. A p ositive answer for co vering cylinder bodies In Section 3 we described examples o f con vex bod ies K and L such th at the orthogonal projections L ξ of L covered the corresponding projections K ξ of K for all ξ ∈ G ( n , k ), e ven though V n ( L ) < V n ( K ). The next theorem shows that this volume anomaly ca n be av oided if L ∈ C n , k . Theor em 6.1. Let K , L ∈ K n and let 1 ≤ k ≤ n − 1 . Suppose that t he orthogonal pr ojection s L ξ of L can cover the corre sponding pr oj ections K ξ of K for all ξ ∈ G ( n , k ) . If L ∈ C n , k , then V n ( K ) ≤ V n ( L ) . If, in addition, the set L has non-empty interior , then V n ( K ) = V n ( L ) if and only if K and L are tr anslates. Pr oof. If t he orthogonal projections L ξ of L can cover the correspond ing projec- tions K ξ of K for all ξ ∈ G ( n , k ), then V n − 1 , 1 ( Q , K ) ≤ V n − 1 , 1 ( Q , L ) for all Q ∈ C n , k , by Theorem 5.3. If L ∈ C n , k as well, then V n − 1 , 1 ( L , K ) ≤ V n − 1 , 1 ( L , L ) = V n ( L ) Meanwhile, the Minkowski m ixed volume inequality (4) asserts that V n ( L ) ( n − 1) / n V n ( K ) 1 / n ≤ V n − 1 , 1 ( L , K ) , Hence V n ( K ) ≤ V n ( L ). If equalit y hol ds and V n ( L ) > 0, then K and L are homo- thetic bo dies of t he same volume by the equality cond itions of (4), so that K and L must be translates.  18 The simplicial countere x amples of Section 2 along with Theorem 6.1 yield the following immediate corollary . Corollary 6.2. An n-dimens ional simplex is ne ver an element of C n , n − 1 . In particular , the coll ection of ( n − 1)-cylinder bodies C n , n − 1 forms a pr oper subset of C n , n = K n . More generally we ha ve the follo wi ng. Corollary 6.3. F or 1 ≤ i < j ≤ n the set C n , i is a pr oper subset of C n , j . Pr oof. It fol lows directly from the definit ion of C n , i that C n , i ⊆ C n , j when i < j . It remains to show that C n , i , C n , j when i < j . T o see thi s, ob serve that the s et L const ructed in the proof of Theorem 3.2 satisfies L ∈ C n , k + 1 , because L is λ -decompo sable for λ = ( k + 1 , 1 , . . . , 1). Let K also be cho sen as in the proof of T heorem 3.2. Recall that the k -shadows L ξ of L can cover those of K for ev ery ξ ∈ G ( n , k ). Since V n ( L ) < V n ( K ), it follows from Theorem 6.1 that L < C n , k . Hence C n , k , C n , k + 1 .  In other words, the collections C n , k form a strictly increasing chain C n , 1 ⊂ C n , 2 ⊂ · · · ⊂ C n , n − 1 ⊂ C n , n = K n where the elements of C n , 1 are precisely the centrally symmetric sets in K n . 7. A geometric inequ ality for cylinder bodies For positive in tegers n ≥ 2 denote σ n =          √ n + 2 2 n + 2 if n is e ven 1 2 √ n if n is odd Recall that we denote the su rface area of a conv ex body K by S ( K ) and the minimal width of K b y d K . Theor em 7.1 (Cylind er body inequality) . Let K ∈ K n . If K ∈ C n , i , then (16) σ i d K S ( K ) ≤ nV n ( K ) . Pr oof. If V n ( K ) = 0 then d K = 0 as well, so that both sides of (16) are zero. Suppose that V n ( K ) > 0. By Steinhagen’ s in equality (10) and the fact that dim K = n , r K ξ ≥ σ i d K ξ ≥ σ i d K , 19 for each subspace ξ ∈ G ( n , i ), where d K ξ is computed from within the subspace ξ . For 0 ≤ ǫ ≤ 1 , denote K ǫ = ǫ K + (1 − ǫ ) σ i d K B n , where B n is an n -dimensional unit Euclidean ball. Since σ i d K B n ⊆ r K ξ B n ⊆ K ξ up to translation, we ha ve K ǫ ξ ⊆ ǫ K ξ + (1 − ǫ ) K ξ = K ξ up to translation, for each subspace ξ ∈ G ( n , i ). If K is an i -c ylinder body , then V n ( K ǫ ) ≤ V n ( K ), b y Theorem 6.1. Moreover , Steiner’ s formula (12) impli es that V n ( K ǫ ) = ǫ n V n ( K ) + ǫ n − 1 (1 − ǫ ) σ i d K S ( K ) + (1 − ǫ ) 2 f ( ǫ ) , where f ( ǫ ) is a polynomial in ǫ . Since V n ( K ǫ ) − V n ( K ) ≤ 0 for 0 ≤ ǫ < 1, we ha ve ( ǫ n − 1) V n ( K ) + ǫ n − 1 (1 − ǫ ) σ i d K S ( K ) + (1 − ǫ ) 2 f ( ǫ ) ≤ 0 so that ǫ n − 1 σ i d K S ( K ) + (1 − ǫ ) f ( ǫ ) ≤ (1 + ǫ + · · · + ǫ n − 1 ) V n ( K ) for all 0 ≤ ǫ < 1. A s ǫ → 1 this impli es that σ i d K S ( K ) ≤ nV n ( K ).  8. A v olu me ra tio bound In Section 2 we described n -dimensional conv ex b odies K and L such that t he orthogonal projection K u can be trans lated inside L u for ev ery direction u , while V n ( K ) > V n ( L ) . In s uch instances, one could ask in stead for an upper bound on the volume ratio V n ( K ) V n ( L ) . An applicati on of Theorem 6 .1 yield s the following crude estimate. Theor em 8.1. Let K , L ∈ K n , and suppose that the orthogonal ( n − 1) -dimension al pr ojection s L u of L can co ver the corr esponding pr ojections K u of K for all dir ec- tions u. Then V n ( K ) ≤ nV n ( L ) . Recall that t he dia meter D K of a con vex body K i s the maximum distance between any two poin ts of th e b ody K , and is also equ al to the maxi mum width , that is, t he maxim um d istance between any two parallel supp orting hyperplanes of K . Pr oof. Suppose that t he diameter D L of L is realized in the uni t direction v . A standard Steiner symmetrization (or , alternati vely , shaking) ar gu ment i mplies that V n ( L ) ≥ 1 n D L V n − 1 ( L v ) . Let ¯ v denote t he unit line se gm ent having endpoints at the origin o and at v , and let C be the orthogon al cylinder in R n giv en b y C = L v ⊕ D L ¯ v . After a sui table translation, we may assume that L ⊆ C . From the orig inal covering assumption 20 for L it then follows t hat each projection K u can be translated inside the corre- sponding projection C u of the cylinder C . By Th eorem 6.1, it then follows that V n ( K ) ≤ V n ( C ) = D L V n − 1 ( L v ) ≤ nV n ( L ) .  9. S ome open questions The re sults of th e pre vious sections motiv ate seve ral open questions about con- vex bodies and projections. I. Let K , L ∈ K n such that V n ( L ) > 0, and let 1 ≤ k ≤ n − 1. Suppose that the orthogonal projections L ξ of L can cover the correspondi ng projectio ns K ξ of K for all ξ ∈ G ( n , k ). What is the best upper bound for the ratio V n ( K ) V n ( L ) ? II. Giv en a partit ion λ of a pos itive integer n , define C λ to be the collection of all con vex bodies that can be approximated by Blaschke sum s of µ -decomposable conv ex bodies, taken over all partitions µ t hat refine t he partition λ . If λ and σ are incomparable partitions of n (wit h respect to partition refinement), how are C λ and C σ related? Can we describe their relative geometric significance in the context of projections? III. Zonoids can be thought of as th e im age of the proj ection bod y operator on con vex s ets or of the cosine transform on s upport functions, and intersection bodies are construct ed by taking the Radon transform of the radial fun ction of a con vex (or star-shaped) set [5, 9, 14]. I s there an analogous integral geometric description for the f ami lies C λ and C n , k ? IV . What simpl e tests, conditio ns, o r i nequalities det ermine whether o r not a con vex body K is an element of some C λ or C n , k ? V . Let K , L ∈ K n such that V n ( L ) > 0, and let 1 ≤ k ≤ n − 1. Suppose that the orthogonal projections L ξ of L can cover the correspondi ng projectio ns K ξ of K for all ξ ∈ G ( n , k ). Under what simple (easy to st ate, easy to verify) additi onal conditions does it follow that K can be translated inside L ? 21 R efere nces [1] T . Bonnesen and W . Fenchel, Theory o f Conve x B odies , BCS Associates, Moscow , Idaho , 1987. [2] H. Busemann and C. M. 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