Functional inequalities derived from the Brunn-Minkowski inequalities for quermassintegrals

F unctional inequalities deriv ed from the Brunn–Mink o wski inequalities for quermassi n tegrals Andrea Colesanti & E u g enia Sao r ´ ın G´ omez ∗ Abstract W e use Brunn–Mink o wski inequalities for quermassin tegrals to deduce a f amily of inequal- ities of P oincar ´ e t yp e on the unit sphere and o n the b oundary of smooth con v ex b o dies in the n –dimensional Euclidean space. AMS 2000 Subje ct Classific ation: 52A20, 26D10 1 In tro d uction The main idea of this pap er is to use Brunn–Mink o wski inequalities for quermassin tegrals to deriv e a family of inequalities of Poinc ar´ e t yp e on the unit sphere and on the b oundar y of con v ex bo dies in the n –dimensional Euclidean space. This t ype of r esearch w as initiated in [4] where the case of the classic Br unn–Minko ws ki inequality is considered. Let K ⊂ R n b e a conv ex b o dy , i.e. a (non–empt y) compact con v ex set. The querm assin te- grals of K , denoted b y W 0 ( K ), W 1 ( K ), . . . , W n ( K ), arise naturally in the p olynomial expression of the v olume of the outer parallel bo dies o f K g iven by t he w ell kno wn Steiner formula : H n ( K + tB ) = n X i =0 t i  n i  W i ( K ) , t ≥ 0 . where B is the unit ball of R n , K + tB = { x + ty : x ∈ K , y ∈ B } is the outer pa rallel b o dy of K at distance t ≥ 0 and H n is the n –dimensional Leb esgue measure. F or a detailed study of quermassin t egr a ls w e refer to [11, § 4.2]. Some of the quermassin tegrals hav e familiar geometric meaning: W 0 ( K ) is the volume (i.e. the Leb esgue measure) of K , while W 1 ( K ) is, up to a dimensional factor, the surface area of K . Each quermassin tegra l W i , i < n , satisfies a ∗ Suppo rted by EU Pro ject Phenomena in High Dimensions MR TN-CT-2 004-5 1195 3. 1 Brunn–Mink owsk i t yp e inequalit y: for ev ery K and L con v ex b o dies and for ev ery t ∈ [0 , 1] w e ha v e W i ((1 − t ) K + tL ) 1 / ( n − i ) ≥ (1 − t ) W i ( K ) 1 / ( n − i ) + tW i ( L ) 1 / ( n − i ) ; (1) for i < n − 1 equalit y holds if and only if K is homothetic to L . When i = 0 this is the classic Brunn–Mink owsk i inequality . In general, the ab ov e inequalities can b e obtained as consequences of the Alek sandro v–F enc hel inequalities (see for instance [11, § 6.4]) . Inequalit y (1) claims that the functional W 1 / ( n − i ) i is conc av e in the class of conv ex b o dies; heuristically , t his implies that the se c ond variation of this functional, whenev er it exists , mus t b e negativ e semi–definite. In this paper w e t r y to make this argumen t more precise and w e study its consequence s. Throughout the pap er w e use the notion o f elemen tary symmetric functions of (the eigen- v alues of ) symmetric matrices. In our notatio n, if A is a N × N r eal symmetric matrix, for r ∈ { 0 , 1 , . . . , N } , S r ( A ) is the r –th elemen tary sy mmetric function of the eigenv alues of A and ( S ij r ( A )) is the r –c ofac tor ma trix of A ; these no tions and their prop erties are recalled in § 2. If K ⊂ R n is a conv ex b o dy of class C 2 + (see § 2 for the definition) then, for i < n , W i ( K ) = c ( n, i ) Z S n − 1 h K S n − i − 1 (( h K ) ij + h K δ ij ) d H n − 1 , (2) where c ( n, i ) is a constant and ( h K ) ij are the second co v ariant deriv ativ es of the supp ort func- tion h K of K (see formu la (5.3.11 ) in [11] for the v alue of c ( n, i ) and § 2 for precise definitions). This integral represen tation f orm ula a llo ws t o compute explicitly the first and second direc- tional deriv ativ es of quermassin tegrals. Then, imp osing the Brunn–Mink o wski inequality (1) w e obtain the follo wing results. Theorem 1. L et K ⊂ R n b e a c o nvex b o dy of class C 2 + , ν b e its Gauss map and I ∈ { 1 , . . . , n − 1 } . F or every ψ ∈ C 1 ( ∂ K ) , if Z ∂ K ψ S I − 1 ( D ν ) d H n − 1 = 0 (3) then I Z ∂ K ψ 2 S I ( D ν ) d H n − 1 ≤ Z ∂ K  ( S ij I ( D ν )) ∇ ψ , ( D ν ) − 1 ∇ ψ  d H n − 1 . (4) Theorem 2. L et h b e the supp ort function of a c on vex b o d y K ⊂ R n of clas s C 2 + and J ∈ { 1 , . . . , n − 1 } . F or every φ ∈ C 1 ( S n − 1 ) , if Z S n − 1 φS J ( h ij + hδ ij ) d H n − 1 = 0 (5) then ( n − J ) Z S n − 1 φ 2 S J − 1 ( h ij + hδ ij ) d H n − 1 ≤ Z S n − 1  ( S ij J ( h ij + hδ ij )) ∇ φ, ∇ φ  d H n − 1 . (6) 2 Theorems 1 and 2 are the tw o faces of the same coin; they can b e obtained one from eac h other by the c hange of v aria ble prov ided by the Gauss ma p. The cases I = 1 of Theorem 1 and J = n − 1 o f Theorem 2 we re already prov ed in [4], as consequences of the classic Brunn–Mink owsk i inequality . Another proo f of Theorems 1 and 2 in t hese sp ecial cases, based on a functional inequality due to Brascamp and Lieb (see [2]), w as commun icated to us by Cordero–Erausquin ([5]). One w a y to lo o k at (3)–(4) and (5)–(6) is as inequalities of P oincar ´ e type, where a weigh ted L 2 –norm of a function is b ounded by a w eigh t ed L 2 –norm of its gradient, under a zero–mean t yp e condition. In particular, c ho osing K = B (the unit ball) in Theorem 1, or equiv alen tly h ≡ 1 in Theorem 2, w e reco ver the usual P o incar ´ e inequalit y on S n − 1 with the optimal constan t : Z S n − 1 φ ( x ) d H n − 1 ( x ) = 0 ⇒ Z S n − 1 φ 2 ( x ) d H n − 1 ( x ) ≤ 1 n − 1 Z S n − 1 |∇ φ ( x ) | 2 d H n − 1 ( x ) . (7) W e also note that inequalities (4) and (6), under side conditions (3) and (5 ) resp ectiv ely , are optimal. This fa ct, prov ed in Remark 1, § 5, is a simple consequence of the in v ariance of quermassin tegrals under translations. When J = 1 we can remo v e t he smo othness assumption on K (or equiv alen tly on h ) in Theorem 2. Indeed w e hav e S J − 1 = S 0 ≡ 1 and S ij 1 ( h ij + hδ ij ) = δ ij . Moreo v er S 1 ( h ij + hδ ij ) d H n − 1 = [∆ h + ( n − 1) h ] d H n − 1 can b e replaced by dA 1 ( K , · ), where A 1 ( K , · ) denotes the ar e a me asur e of or der one of K (see § 5 for the definition). Theorem 3. L et K ⊂ R n b e a c onvex b o dy and let A 1 ( K , · ) b e its ar e a me asur e of or d e r on e. F or ev e ry φ ∈ C 1 ( S n − 1 ) , if Z S n − 1 φ ( x ) dA 1 ( K , x ) = 0 , (8) then Z S n − 1 φ 2 ( x ) d H n − 1 ( x ) ≤ 1 n − 1 Z S n − 1 |∇ φ ( x ) | 2 d H n − 1 ( x ) . Hence Theorem 3 extends the usual P oincar´ e inequalit y (7 ) on S n − 1 when the zero–mean condition is r eplaced b y (8). F or n = 2 this leads to an extension of the we ll kno wn Wirtinger ine quality , stated in Corollary 1 of § 5 . In higher dimension Theorem 3 to g ether with some recen t dev elopmen ts on the Christoffel problem ([7], [10]) leads to the following result. Theorem 4. L et K ⊂ R n b e a c onvex b o dy c o n taining the origin in its interior, such that Z S n − 1 xρ K ( x ) d H n − 1 ( x ) = 0 , (9) wher e ρ K is the r adial function of K . Th en, for every φ ∈ C 1 ( S n − 1 ) , Z S n − 1 φ ( x ) ρ K ( x ) d H n − 1 ( x ) = 0 ⇒ Z S n − 1 φ 2 ( x ) d H n − 1 ( x ) ≤ 1 n − 1 Z S n − 1 |∇ φ ( x ) | 2 d H n − 1 ( x ) . 3 Note that condition (9) is fulfilled when K is cen trally symmetric. Ac kno wledgmen t . W e would lik e t o tha nk L. Al ´ ıas Linares for his precious help in the pro of of Lem ma 1. 2 Preliminaries 2.1 Elemen tary symmetric functions Let N b e an integer; for a N × N symmetric matrix A = ( a ij ) ha ving eigenv alues λ 1 , . . . , λ N , and fo r k ∈ { 0 , 1 , . . . , N } w e define the k –th elemen tary symmetric function of the eigen v alues of A as follo ws S k ( A ) = X 1 ≤ i 1 < ··· 0 on S n − 1 } consists of supp ort functions of conv ex b o dies of class C 2 + . When K is of class C 2 + , the quermassin t egr a ls of K can b e ex pressed as in tegrals in v olving the sup p ort function h K of K . In fact, for i ∈ { 0 , 1 , . . . , n − 1 } , W i ( K ) = 1 n  n − 1 n − i − 1  − 1 Z S n − 1 h K S n − i − 1 (Ξ − 1 ) d H n − 1 (13) 5 (see form ula (5.3.11) in [11]). Note that f or K, L ∈ K n and t ∈ [0 , 1] we hav e h (1 − t ) K + tL = (1 − t ) h K + th L . F rom the ab o v e f acts and inequalit y (1) w e deduce the following result. Prop osition 2. F or i ∈ { 0 , 1 , . . . , n − 1 } define the functional F i : C → R + , F i ( h ) = Z S n − 1 h S n − i − 1 (Ξ − 1 ) d H n − 1 . Then ( F i ) 1 / ( n − i ) is c onc ave in C . 3 A lemma con cerning Hessian op erato rs on the sphe re This section is dev oted to pro v e the follo wing result, whic h will b e used in the pro ofs of Theorems 1 a nd 2. Lemma 1. L et u ∈ C 2 ( S n − 1 ) , k ∈ { 1 , . . . , n − 1 } and let { E 1 , . . . E n − 1 } b e a lo c al orthonormal fr ame of v e ctor fields on S n − 1 . Then, fo r every i ∈ { 1 , . . . , n − 1 } , div j ( S ij k ( ∇ 2 u + u I )) := n − 1 X j =1 ∂ ∂ E j S ij k ( ∇ 2 u + u I ) = 0 , wher e ∂ ∂ E j denotes the c ov ariant differ ential acting on E j and I den otes the ( n − 1) × ( n − 1) identity matrix. The case k = n − 1 of the prece ding lemma w as pro v ed b y Cheng and Y au in [3] (see pag e 504). W e also note t ha t a n analogous result is v alid in the Euclidean setting, with ( ∇ 2 u + uI ) replaced by ∇ 2 u (see for instance [8, Prop osition 2 .1] and [9 , § 2.3]). Our pro of f o llo ws the argumen t of [9] f or the Euclidean case and uses some standard to ols from differen tial geometry on S n − 1 . Pr o of. F o r k ∈ { 0 , 1 , . . . , N } , the k –th eleme n tary symmetric functions o f a symmetric N × N matrix A = ( a ij ) can b e written in the follo wing w a y (see, for instance, [8]) S k ( A ) = 1 k X δ  i 1 , . . . , i k j 1 , . . . , j k  a i 1 j 1 · · · a i k j k where the sum is taken ov er all p ossible indices i r , j r ∈ { 1 , . . . , N } for r = 1 , . . . , k and the Kro- nec ke r sym b ol δ  i 1 ,...,i k j 1 ,...,j k  equals 1 (resp ective ly , − 1) when i 1 , . . . , i k are distinct and ( j 1 , . . . , j k ) 6 is an eve n (resp ective ly , o dd) p erm utation of ( i 1 , . . . , i k ); otherwise it is 0 . Using the ab o v e equalit y we hav e S ij k ( A ) = 1 ( k − 1 ) ! X δ  i, i 1 , . . . , i k − 1 j, j 1 , . . . , j k − 1  a i 1 j 1 · · · a i k − 1 j k − 1 . Hence w e can write ( k − 1 ) ! n − 1 X j =1 ∂ ∂ E j S ij k ( ∇ 2 u + u I ) = (14) = n − 1 X j =1 X δ  i, i 1 , . . . , i k − 1 j, j 1 , . . . , j k − 1  ∂ ∂ E j (( u i 1 j 1 + uδ i 1 j 1 ) · · · ( u i k − 1 j k − 1 + uδ i k − 1 j k − 1 )) = n − 1 X j =1 X δ  i, i 1 , . . . , i k − 1 j, j 1 , . . . , j k − 1  [( u i 1 j 1 j + u j δ i 1 j 1 )( u i 2 j 2 + uδ i 2 j 2 ) · · · ( u j k − 1 i k − 1 + uδ i k − 1 j k − 1 ) + · · · + ( u i 1 j 1 + uδ i 1 j 1 )( u i 2 j 2 + uδ i 2 j 2 ) · · · ( u i k − 1 j k − 1 j + u j δ i k − 1 j k − 1 )] . In the last sum, for fixed i 1 , . . . i k − 1 , j 1 , . . . j k − 1 , j , let us consider the terms A = δ 1 ( u i 1 j 1 j + u j δ i 1 j 1 ) C and B = δ 2 ( u i 1 j j 1 + u j 1 δ i 1 j 1 ) C , where δ 1 = δ  i, i 1 , i 2 , . . . , i k − 1 j, j 1 , j 2 , . . . , j k − 1  , δ 2 = δ  i, i 1 , i 2 , . . . , i k − 1 j 1 , j, j 2 , . . . , j k − 1  , and C = ( u i 2 j 2 + uδ i 2 j 2 ) · · · ( u j k − 1 i k − 1 + uδ i k − 1 j k − 1 ) . Clearly δ 1 = − δ 2 . Moreo v er we hav e the follo wing relation concerning co v ariant deriv atives on S n − 1 (see, for instance, [3]) u r st + u t δ r s = u r ts + u s δ r t , ∀ r, s, t = 1 , · · · , n − 1 . Hence A + B = 0 . W e ha v e pro v ed that to t he term A in the last sum in (14) it corresp onds another term B , uniquely determined, whic h cancels out with A . The same argumen t can b e rep eated for any ot her term o f the sum and this concludes the pro of. 4 Pro of o f The orems 1 and 2 In this section K is a fixed con v ex bo dy of class C 2 + and h is its supp ort function; in particular h ∈ C . W e recall that Ξ − 1 = ( h ij + hδ ij ) and, for k ∈ { 0 , . . . , n − 1 } , F k ( h ) = Z S n − 1 h S n − k − 1 (Ξ − 1 ) d H n − 1 . 7 Note that if φ ∈ C ∞ ( S n − 1 ) and ǫ is sufficien tly small, then h + sφ ∈ C for | s | ≤ ǫ . W e will denote b y Ξ − 1 s the matrix (( h s ) ij + h s δ ij ). Prop osition 3. L et k ∈ { 0 , . . . , n − 1 } , h ∈ C , φ ∈ C ∞ ( S n − 1 ) and ǫ > 0 b e such that h s = h + sφ ∈ C for e v ery s ∈ ( − ǫ, ǫ ) . L et f ( s ) = F k ( h s ) . Then f ′ ( s ) = ( n − k ) Z S n − 1 φ S n − k − 1 (Ξ − 1 s ) d H n − 1 , s ∈ ( − ǫ, ǫ ) . Pr o of. f ′ ( s ) = Z S n − 1 ∂ ∂ s h s S n − k − 1 (Ξ − 1 s ) d H n − 1 = Z S n − 1  φ S n − k − 1 (Ξ − 1 s ) + h s ∂ ∂ s ( S n − k − 1 (Ξ − 1 s ))  d H n − 1 (15) = Z S n − 1 " φ S n − k − 1 (Ξ − 1 s ) + h s n − 1 X i,j =1 S ij n − k − 1 (Ξ − 1 s )( φ ij + φδ ij ) # d H n − 1 . In tegrating b y parts t wice and using Lemma 1 w e obtain Z S n − 1 h s n − 1 X i,j =1 S ij n − k − 1 (Ξ − 1 s ) φ ij d H n − 1 = Z S n − 1 φ n − 1 X i,j =1 S ij n − k − 1 (Ξ − 1 s )( h s ) ij d H n − 1 . (16) On the other hand, by ( 1 1) n − 1 X i,j =1 S ij n − k − 1 (Ξ − 1 s )(( h s ) ij + h s δ ij ) = ( n − k − 1) S n − k − 1 (Ξ − 1 s ) . (17) The pro of is completed inserting (16) a nd (17) in (15). The pro of of the next result a s traightforw ard consequence of Prop osition 3. Prop osition 4. In the assumptions and notations o f Pr op osition 3 f ′′ (0) = ( n − k ) Z S n − 1 φ n − 1 X i,j =1 S ij n − k − 1 (Ξ − 1 )( φ ij + φδ ij ) d H n − 1 . (18) W e are now ready to prov e Theorems 1 and 2; w e b egin with the latter. 8 Pr o of o f The or em 2. Without loss of generalit y w e ma y assume that φ ∈ C ∞ ( S n − 1 ). Fix ǫ > 0 suc h t hat h + sφ ∈ C for s ∈ ( − ǫ, ǫ ) a nd let k = n − J − 1. As a b ov e, w e set f ( s ) = F k ( h + sφ ) and define g ( s ) = f 1 n − k ( s ). W e kno w from Prop osition 2 that g is a conca v e function and so g ′′ (0) = 1 n − k  1 n − k − 1  f (0) 1 n − k − 2 ( f ′ (0)) 2 + ( f (0)) 1 n − k − 1 f ′′ (0)  ≤ 0 . Notice that, b y Prop o sition 1, the assumption (5) giv es exactly f ′ (0) = 0, so the condition g ′′ (0) ≤ 0 b ecomes ( f (0) ) 1 I − 1 f ′′ (0) ≤ 0. Since f (0) = W k ( K ) > 0 it follows f ′′ (0) ≤ 0 . Now (18) giv es us Z S n − 1 φ 2 n − 1 X i,j =1 S ij J (Ξ − 1 ) δ ij d H n − 1 ≤ − Z S n − 1 φ n − 1 X i,j =1 S ij J (Ξ − 1 ) φ ij d H n − 1 . In tegrating b y parts in the righ t hand–side and using Lemma 1 w e obtain Z S n − 1 φ n − 1 X i,j =1 S ij J (Ξ − 1 ) φ ij d H n − 1 = − Z S n − 1 n − 1 X i,j =1 S ij J (Ξ − 1 ) φ i φ j d H n − 1 and w e are done with the aid of part v) of Prop o sition 1. F or the pro of of Theorem 1 w e need the following auxiliary result. Lemma 2. L et φ ∈ C ∞ ( S n − 1 ) a n d ψ ( x ) = φ ( ν ( x )) , x ∈ ∂ K , wher e ν is the Gauss m ap of K . Fix r ∈ { 1 , . . . , n − 1 } . Then for eve ry y ∈ S n − 1 1 det(Ξ − 1 ( y ))  ( S ij r (Ξ − 1 ( y ))) ∇ φ ( y ) , ∇ φ ( y )  =  (( D ν ( x )) − 1 ( ∇ ψ ( x )) , S ij n − r (Ξ( x )) ∇ ψ ( x ))  , wher e x = ν − 1 ( y ) and Ξ( x ) = D ν ( x ) . Pr o of. W e ma y assume tha t Ξ − 1 ( y ) is diagonal: Ξ − 1 ( y ) = diag ( λ 1 , . . . , λ n − 1 ) , λ i > 0 , i = 1 , . . . , n − 1 . Then D ν ( x ) = diag( µ 1 , . . . , µ n − 1 ) , µ i = 1 λ i , i = 1 , . . . , n − 1 . In particular ∇ ψ ( x ) = D ν ( x ) ∇ φ ( ν ( x )) = n − 1 X i =1 µ i φ i ( y ) . (19) 9 By Proposition 1 the matrix ( S ij r (Ξ − 1 ( y ))) is also diagonal and its eigen v alues are giv en b y Λ s = S r − 1 (diag( λ 1 , . . . , λ s − 1 , λ s +1 , . . . , λ n − 1 )) , s = 1 , . . . , n − 1 . Using again Prop osition 1 w e get P n − 1 i,j =1 S ij r (Ξ − 1 ( y )) φ i ( y ) φ j ( y ) det(Ξ − 1 ( y )) = n − 1 X i =1 Λ i det(Ξ − 1 ) φ 2 i ( y ) = n − 1 X i =1 µ i S n − r − 1 (diag( µ 1 , . . . , µ i − 1 , µ i +1 , . . . , µ n − 1 )) φ 2 i ( y ) = n − 1 X i =1 µ i S ii n − r ( D ν ( x )) φ 2 i ( y ) = h∇ ψ ( x ) , ( S ij n − r ( D ν ( x ))) ∇ φ ( y ) i . The conclusion of the lemma follow s from the first equalit y in (1 9) and the symmetry of the matrix ( S ij n − r ( D ν ( x ))). Pr o of o f The or em 1. W e set φ ( y ) = ψ ( ν − 1 ( y )), y ∈ S n − 1 . Consider t he map ν − 1 : S n − 1 → ∂ K ; its Jacobian is giv en by det( D ( ν − 1 )( y )) = det(Ξ − 1 ( y )) > 0 , ∀ y ∈ S n − 1 . Moreo v er, by Prop osition 1 w e ha ve that for ev ery r ∈ { 0 , 1 , . . . , n − 1 } , S r ( D ν ( ν − 1 ( y ))) = S n − r − 1 (Ξ − 1 ( y )) det(Ξ − 1 ( y )) , ∀ y ∈ S n − 1 . Hence w e can write Z ∂ K ψ S I − 1 ( D ν ) d H n − 1 = Z S n − 1 φS n − I (Ξ − 1 ) d H n − 1 , Z ∂ K ψ 2 S I ( D ν ) d H n − 1 = Z S n − 1 φ 2 S n − I − 1 (Ξ − 1 ) d H n − 1 . And, b y Lemma 2, Z ∂ K  S ij I ( D ν ) ∇ ψ, ( D ν ) − 1 ∇ ψ  d H n − 1 = Z S n − 1  ( S ij n − I (Ξ − 1 )) ∇ φ, ∇ φ  d H n − 1 . The pro of is completed applying Theorem 2 with J = n − I . 10 Remark 1. With the notation of the pro of of Theorem 2, let φ ( y ) = h y 0 , y i , where y 0 ∈ S n − 1 is fix ed. Note that condition (5) is verifie d as Z S n − 1 y S J ( h ij ( y ) + hδ ij ( y )) d H n − 1 = Z S n − 1 y dA J ( K , y ) , where A J ( K , · ) is the J –th area measure of K (see [11] or the next section for the definition), and the latter in tegral is ze ro by standard prop erties of area measures. Moreov er, for ev ery s , h + sφ is the supp ort function of a translate o f K . Since q uermassin tegrals are inv ariant with resp ect to translations, the function f is constant in particular f ′′ (0) = 0. This prov es that if φ is as ab o v e w e hav e equalit y in (6). Analogously , c ho osing ψ ( x ) = h x 0 , ν ( x ) i where 0 6 = x 0 ∈ R n is fix ed, w e s ee that condition ( 3) of Theorem 1 is fulfilled and (4) b ecomes an equalit y . 5 The case J = 1 : the p ro of of Theorems 3 and 4 W e start this section recalling the definition of area measures; for a detailed presen tatio n of this topic w e refer the reader to [11, Chapter 5 ]. If K 1 , . . . , K m , m ∈ N , are con vex b o dies in R n and λ 1 , . . . , λ m are non– nega t ive r eal n um b ers, then w e ha v e: H n ( λ 1 K 1 + · · · + λ m K m ) = m X i 1 ,...,i n =1 λ i 1 · · · λ i n V ( K i 1 , . . . , K i n ) . The coefficien ts of the p olynomial at the right hand–side a re called mixe d volumes . Moreo v er, if w e fix ( n − 1) con v ex b o dies K 2 , . . . , K n , there exists a unique non–negativ e Borel measure A ( K 2 , . . . , K n , · ) (called mix e d ar e a me asur e ) su c h that for eve ry con v ex b o dy K 1 V ( K 1 , K 2 , . . . , K n ) = Z S n − 1 h K 1 ( x ) dA ( K 2 , . . . , K n , x ) . F or j = 1 , . . . , n − 1, the ar e a me asur e of order j of a conv ex b o dy K is obtained in the follo wing wa y: A j ( K , · ) = A ( K , . . . , K , B , . . . , B , · ), where K is repeat ed j times and B is the unit ball in R n . An a lt ernativ e definition of area measures is based on a lo cal ve rsion of the Steiner formula (see [1 1, Chapter 4]). In particular, the area measure of order one of K is A 1 ( K , · ) = A ( K , B . . . , B , · ). If K is o f class C 2 + , then it can be pro ved that dA 1 ( K , x ) = 1 n − 1 S 1 (( h K ) ij ( x ) + h K ( x ) δ ij ) d H n − 1 ( x ) . (20) Hence condition ( 5) is equiv a len t to (8) when h is the supp o r t function of a conv ex b o dy of class C 2 + . 11 Pr o of o f The or em 3. W e may a ssume that φ ∈ C ∞ ( S n − 1 ). K can b e approxim ated b y a se- quence K r , r ∈ N , suc h that f or ev ery r , K r is of class C 2 + and ( K r ) r ∈ N con v erges to K in the Hausdorff metric a s r tends to infinity . Fix r ∈ N and let h r b e the supp or t function of K r . F or s sufficien tly small in absolute v alue, consider the function f r ( s ) = Z S n − 1 ( h r + sφ ) S 1 (( h r + sφ ) ij + ( h r + sφ ) δ ij ) d H n − 1 . By Prop osition 2, √ f r is conca ve so that 2 f r (0) f ′′ r (0) − ( f ′ r (0)) 2 ≤ 0. Using (13), Prop ositions 3 a nd 4 (with k = n − 2) and the relation ( S ij 1 ) = ( δ ij ), w e obtain 2 n n − 2 W n − 2 ( K r ) Z S n − 1 φ ( n − 1) φ + n − 1 X i =1 φ ii ! d H n − 1 ≤  Z S n − 1 φS 1 (( h r ) ij + h r δ ij ) d H n − 1  2 . (21) F rom (20) w e kno w that Z S n − 1 φS 1 (( h r ) ij + h r δ ij ) d H n − 1 = ( n − 1) Z S n − 1 φ ( x ) dA 1 ( K r , x ) , where A 1 ( K r , · ) is the first area measure of K r . Moreov er, as r tends to infinit y the seq uence of me asures A 1 ( K r , · ) con ve rges w eakly to A 1 ( K , · ) (se e [11, The orem 4.2.1]). This implies lim r →∞ Z S n − 1 φ ( x ) dA 1 ( K r , x ) = Z S n − 1 φ ( x ) dA 1 ( K , x ) = 0 . (22) On the other hand W n − 2 ( K r ) con v erges t o W n − 2 ( K ) as r tends to infinit y (by standa r d con ti- n uit y results on quermassin tegra ls) and W n − 2 ( K ) > 0 as K has interior p o ints. The conclusion follo ws letting r → ∞ in (21), using (22) and in tegrating b y parts. As men tioned in the In t r o duction, Theorem 3 extends the usual (sharp) P oincar ´ e inequalit y (7) on S n − 1 when the us ual zero–mean condition is replaced b y (8). Clearly , in order to apply this result it w ould b e useful to understand when a measure µ o n S n − 1 is t he area measure of order o ne of some con v ex b o dy . This amoun ts to solv e the Christoffel problem for µ (see for instance [11, § 4.3]). F or n = 2 this problem coincides with the Mink ow ski problem and its solution is completely understoo d. Let µ b e a non–negativ e Borel measure on S 1 suc h that: i) µ is not the sum of tw o p oint–mass es; ii) Z S 1 x dµ ( x ) = 0 . Then there exists a con v ex b o dy K in R 2 suc h that A 1 ( K , · ) = µ ( · ) (note that conditions i) and ii) are also necess ary in order t ha t µ is the a r ea measure of order one of some con vex b o dy). Hence w e ha ve the follo wing extension of the w ell kno wn Wirtinger ine quality . 12 Corollary 1. L et ν b e a non –ne gative Bor el me asur e on [0 , 2 π ] such that ν is not the sum of two p oint–m a s ses and Z 2 π 0 sin θ dν ( θ ) = Z 2 π 0 cos θ dν ( θ ) = 0 . Then, for every φ ∈ C 1 ([0 , 2 π ]) such that φ (0) = φ (2 π ) Z 2 π 0 φ ( θ ) d ν ( θ ) = 0 ⇒ Z 2 π 0 ( φ ( θ )) 2 dθ ≤ Z 2 π 0 ( φ ′ ( θ )) 2 dθ . In higher dimension the Christoffel problem is more complicated. Necessary and sufficien t conditions for a measure µ to b e the first area measure of some conv ex b o dy w ere found by Firey [6] and Berg [1] (see a lso [11, § 4.2]). On the o ther hand these conditions are not easy to use in practice. A considerable progress (in a larger class of problems) has b een made b y Guan and Ma in [7] and Sheng, T rudinger and W ang in [10] where a rather simple sufficien t condition is found. Here w e state this result in the case of area measures of o r der one. Theorem 5 ( Guan, Ma, Sheng, T rudinger, W ang). L et f ∈ C 1 , 1 ( S n − 1 ) , f > 0 and let g = 1 /f . If Z S n − 1 xf ( x ) d H n − 1 ( x ) = 0 , and the m atrix ( g ij + g δ ij ) i s p ositive sem i – definite a.e. on S n − 1 , then ther e exists a c onvex b o dy L , uniquely determine d up to tr ansl a tion s, such that dA 1 ( L, · ) = f ( · ) d H n − 1 ( · ) , i.e. f is the density of S 1 ( K , · ) with r esp e ct to H n − 1 ( · ) . Using the ab ov e result and Theorem 3, w e no w proceed to sho w Theorem 4 . Pr o of o f The or em 4. W e recall that the radial function ρ K of K is defined as ρ K ( x ) = max { λ ≥ 0 | λx ∈ K } . Let H b e the polar b o dy of K : H = { x ∈ R n : h x, y i ≤ 1 , ∀ y ∈ K } . H is still a con v ex b o dy and the or ig in belongs to its in terior. Note that (see for instance [11, Remark 1.7.7]) ρ K = 1 h H , on S n − 1 . Let H r , r ∈ N , b e a sequence o f con ve x b o dies con v erging to H in the Hausdorff metric as r tends to infinit y , suc h that each H r is of class C 2 + . By h yp othesis (9) we may assume that Z S n − 1 x 1 h H r ( x ) d H n − 1 ( x ) = 0 ∀ r ∈ N . 13 Setting h r = h H r w e ha v e that h r → h H uniformly on S n − 1 and (( h r ) ij + h r δ ij ) > 0 on S n − 1 for ev ery r ∈ N . (23) Hence for ev ery r ∈ N w e can apply Theorem 5 with f = f r = 1 /h r , obtaining a con vex b o dy L r suc h that dA 1 ( L r , · ) = f r ( · ) d H n − 1 ( · ) . As H is a conv ex b o dy with in terior p oints , we ha v e that c < h H < C on S n − 1 , for suitable p ositiv e constan ts c and C . Using the uniform con v ergence we obtain that there exist d, D > 0 suc h t hat d ≤ f r ( x ) ≤ D , ∀ x ∈ S n − 1 , ∀ r ∈ N . Hence w e may apply Lemma 3.1 in [7] to deduce that the sequence L r is b ounded and by the Blasc hke selection theorem (see [1 1 , Theorem 1.8.6]), up to a subsequence , it conv erges to a conv ex b o dy L in the Hausdorff metric. As already noticed in the pro of of Theorem 3, the seque nce of measures A 1 ( L r , · ) con v erges weakly to A 1 ( L, · ) as r tends to infinit y . Consequen tly dA 1 ( L, · ) = 1 h H ( · ) d H n − 1 ( · ) = ρ K ( · ) d H n − 1 ( · ) . The conclusion follows applying Theorem 3. References [1] Ch. Be rg , Corps con v exes et p oten tiels sph ´ erique, D ansk e Vid. Selsk ab. Mat.–fys. Medd. 37 (1969), 6 . [2] H. Brascamp & E. Lieb , On extensions of the Brunn–ink o wski and Pr´ ek opa– Leindler inequalit y , including inequalities fo r log conca ve f unctions, and with an application to diffusion equation, J. F unct. Anal. 22 (1976), 366–389. [3] S. T. Cheng & S. T. Y au , O n the regularity of the solution of the n –dimensional Mink owsk i problem, Commun . Pure Appl. Math. 29 (1976), 4 95–516. [4] A. Colesanti , F rom the Brunn-Mink ow ski inequalit y to a class of P oincar ´ e t yp e inequal- ities, to app ear on Comm unications in Con temp or a ry Mathematics . [5] D . Cordero–Era usquin , p ersonal comm unication. [6] W. J. Firey , Christoffel’s problem for general conv ex b o dies, Mathematik a 15 (1968), 7–21. [7] P. Guan, X.- N. M a , The Christoffel–Mink ow ski problem I: con v exit y of solutions of a Hessian eq uation, Inv en t . Math. 151 (2003), 553–577. 14 [8] R. C. Reill y , On the Hessian of a f unction and the curv atures o f its graph, Mic higa n Math. J. 20 (1973), 373–383. [9] P. Salani , Equazioni Hessian e e k − c onvessit` a . Ph.D. Thesis D issertation. Univ ersit` a di Firenze, 1997. [10] W. Sheng, N. Tr udinger, X. -J. W ang , Conv ex hypersurfaces of prescriv ed W ein- garten curv a t ures, Comm. Analysis and Geometry 12 (2004), 2 13–232. [11] R . Schneide r , Convex Bo dies: The Brunn–Minkowski The ory , Cam bridge Univ ersit y Press, Cam bridge, 1993 . Andrea Colesanti Dipartimen to di Matematica ’U. Dini’ – Univ ersit` a di Firenze Viale Morgagni 67/a 50134 F irenze, Italy colesant@ma th.unifi.it Eugenia Saor ´ ın G ´ omez Departamen to de Matematicas – Univ ersidad de Murcia Campus de Espinardo 30100 Murcia, Spain esaorin@um. es 15