Signed Chord Length Distribution. I

Signed Chord Length Distribution P ar t I Ale xander ⁀ Yu. Vlaso v Abstract In this pap er is discussed an applicatio n of signed measures (charges) to description of segmen t and chord l ength distributions in noncon vex bo dies. The signed distribution ma y naturally app ears due to definition via d eriv ativ es of nonnegative auto correlation function simply related with distances distribution b etw een pairs of p oin ts in the b ody . In the w ork is suggested constructive geometri cal in terpretation of such deri va tives and illustrated app ear- ance of “positive” and “negative” elements similar with usual Hanh –Jor dan d ecomposition in measure theory . The construction is also close related with applications of Dirac metho d of chords. Contents 1 Introduction 2 2 Con vex body 3 2.1 Basic geometrical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Dirac’ s m ethod of chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Nonconv ex body 6 3.1 Formal inte gration by par ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Radii (signed) density function . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Chord length (signed) density function . . . . . . . . . . . . . . . . . . . . . . . 8 4 Nonuniform case 11 5 Applications to arbitrary paths 12 A Calculation of distrib ution s for con vex body 14 A-1 Distribution of distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 A-2 A utocorrelation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 A-3 Distribution of radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 A-4 A utocorrelation function and distribution of radii . . . . . . . . . . . . . . . . . . . 1 7 A-5 Chord length distributi on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 B Some equations f or nonuniform case 20 B-1 Distance between points ∆ r ,r ′ = | r ′ − r | . . . . . . . . . . . . . . . . . . . . . 21 B-2 “Optical” length ∆ r ,r ′ = O r ′ r . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 C A verage path length 23 c  A. ⁀ Yu.Vlasov, 2007 1 Signed CLD 1 INTRODUCTION Different prop erties of chord length distribution (CLD) for nonconv ex b odies were discuss e d re- cently in few publications [1, 2, 3, 4, 5, 6, 7]. Let us recall t hree different w ays to introduce CLD for nonconv ex bo dy . Straight line may intersect nonco n vex b ody mor e than one time (see Fig . 1) and we ca n either consider each segment of such line as separate chord or ca lculate sum of all such segments. These tw o metho ds are known as multi-c hord and one-chord distribution (MCD a nd OCD) resp ectively [1, 2, 3]. A B C D A B C D a) b) Figure 1: Nonco n vex bo dies: a ) simply co nnected b) with hole. F or the co n vex c a se pro ba bilit y density function for CLD is prop ortional to second deriv ative of auto correla tion function [1, 4, 5, 6 ]. Suc h a pr operty may be used for a third definition of “the generalized chord distributio n” [4, 5, 6], but stra igh tforward calculations demo nstrates p ossibility of negativity o f such function for some nonco n vex bo dies [6]. Let us ca ll this function her e signe d chor d (length) distribution to av oid some a m biguity of ter m “genera lized” and to emphasize the basic distinguishing prop erty of this function. Definition of CLD for conv ex bo dy has sta ndard pr ob abilistic interpr etation in theor y of ge o- metric pr ob ability and r andom sets [8, 9, 10]. The MCD and OCD cas e s fo r nonconvex b o dy may be describ ed a s well [1, 2, 3]. Is it po ssible to co nsider s imilar p ossibility for t he signe d chor d distribution ? In wonderful ess ay “Ne gative pr ob ability” [11] F eynman wrote that, unlike “ final probability of a verifiable physical even t”, “conditional pro babilities a nd pro babilities of imagined intermediate states may b e negative” and so: “If a physic al the ory for c alculating pr ob abilities yields a ne gative pr ob ability for a given situation under c ertain assume d c onditions, we ne e d not c onclude the the ory is inc orr e ct.” In this review F eynman provided a few exa mples with app earance a nd interpretation of negative proba bilities b oth for quantum and cla s sical physical mo dels. Mathematical extensio n of the measur e theor y for such a pur poses may use so- called signe d me asure s (char ges) [12]. Usually such extension is reduced to standard po sitiv e measures due to Hahn and Jor dan de c omp ositions , corres ponding to expr ession of charge a s difference of tw o po sitiv e measure s [1 2]. In many pro cesses with signed dis tributions the Ha hn dec o mposition, i.e. , splitting of spa c e of event s on po sitiv e and negative parts is quite obvious, e.g. , in simplest examples we have tw o kinds o f even ts: putting and removing ob jects [1 1]. A distinction of the signe d chor d distribution is app earance of negativit y due to differen tiations of positive function without such a natural decomp osition o n p ositive and nega tive elements. Nonconv ex bo dy Fig . 1b may be r e presen ted as a co n vex bo dy and a co n vex hole and it provides some intuitiv e justifica tion of p ossibility to express some distributions using forma l difference of conv ex hull and the hole. Rigor considera tion is more difficult, espec ia lly for chord distribution expressed via seco nd deriv ative, e.g. , metho d derived b elo w in Sec. 3.3 reduces exa mples like Fig. 1 to six “signed” interv als: four “ positive”: [ AD ], [ AB ], [ C D ], [ B C ] and tw o “ ne g ativ e”: [ AC ], [ B D ]. c  A. ⁀ Yu.Vlasov, 200 7 2 Si gned CLD The co n vex case is r e visited in Sec. 2 a nd App endix A. The constr uction of signed c hor d length distribution for nonconv ex b o dy is descr ibed in Sec. 3 . Some implica tions to description of a rbitrary b odies with nonuniform de ns it y are briefly mentioned for c o mpleteness in Sec. 4 and Appendix B. Other ex tens ions, like p olygonal tra jectories ar e a ffected very shortly in Sec. 5 and Appendix C. 2 CONVEX BOD Y 2.1 Basic geometrical models There are ma n y different functions and relations b et ween them used for description pro p erties o f conv ex bo dies [1, 2, 3, 4 , 5, 6, 7, 8 , 9, 1 0, 14, 15, 16]. In present pap er are considered three different kinds o f distributions: distances b et ween p oints Fig. 2a, le ngths of radii (seg men ts) Fig. 2 b, and lengths of chords Fig . 2c. It may b e useful to describ e precisely mo dels of genera tion of each distribution, to avoid some proble ms with a m biguity , similar with widely known Bertrand par ado x [8, 13]. Definition 1. Distanc es distribution function is F η ( l ) = R l 0 η ( x ) dx with density η ( l ) . The distanc es ar e define d by p airs of p oints inside of the b o dy V . Both p oints ar e fr om indep endent uniform distributions, Fig. 2a. Definition 2. R adii distribution function is F ι ( l ) = R l 0 ι ( x ) dx with density ι ( l ) . Th e r adii ar e define d as se gments of r ays fr om a p oint inside of t he b o dy V to the surfac e. The p oints ar e fr om uniform distribution and dir e ct ions of t he r ays ar e isotr opic, Fig. 2 b. Definition 3. Chor d lengths distribution fu n ction is F µ ( l ) = R l 0 µ ( x ) dx with density µ ( l ) . The chor ds ar e define d by interse ction of the b o dy V with isotr opic uniform distribution of lines, Fig. 2 c. a) b) c) Figure 2: Distributions: a ) Distances b etw een p oin ts. b) Radii. c) Chords. It is also conv enient to use au t o c orr elation function γ ( l ) related with distances distribution for three-dimenional b o dy a s η ( l ) = 4 πl 2 γ ( l ) /V , (A8 in Appendix A) where V is volume of V . T he expressio n for auto corre la tion f unction for bo dy with arbitr ary densit y Eq. (A5) to g ether with deriv ation of Eq . (A8) for constant density is reco lle cted for c o mpleteness below in App endix A-2. There is remark able corres p ondence b etw een these densities [1, 2, 3, 4, 5, 6, 15, 16]: 1 h l i µ ( l ) = − ι ′ ( l ) = γ ′′ ( l ) , (2.1) c  A. ⁀ Yu.Vlasov, 200 7 3 Si gned CLD where av erag e chord length h l i = R ∞ 0 l µ ( l ) dl may b e expresse d via volume V and surface a r ea S using a rela tion for three-dimensional conv ex b ody der iv ed in XIX century by Cauch y , Czub er and rediscov er e d later by Dirac et al [8, 10, 14, 1 5] h l i = 4 V S . (2.2) Distribution of lengths o f radii [16] is also known as interior sour c e r andomness [1 5]. Rela tion betw een µ ( l ) and ι ( l ) in Eq. (2.1) is often r epresent ed in integral fo rm [15, 1 6] ι ( l ) = h l i − 1 Z ∞ l µ ( x ) dx = h l i − 1  1 − F µ ( l )  . Prop ortionality b et ween µ ( l ) and seco nd deriv ative o f auto correla tion function γ ( l ) in Eq . (2.1) is also well known and widely used [1, 4, 5, 6]. Here is conv enient for completeness o f pres en tation a nd further explana tio n of nonconv ex ca se to derive all eq ualities in E q. (2.1), cons ide r ing µ ( l ), ι ( l ), γ ( l ) and η ( l ) as generaliz e d functions. The usual definition of gener alize d function [12] is a c ontinuous line ar fu n ctional T ( φ ) on a space of a test functions φ . F or an integrable function ψ the functiona l T ψ is defined for a test function φ ( x ) as T ψ ( φ ) = Z ∞ −∞ ψ ( x ) φ ( x ) dx. (2.3) The gener alize d derivative [1 2] is defined a s functional T ′ ( φ ) ≡ − T ( φ ′ ) . (2.4) Such a definition ensures der iv ative of any order for T ψ with ar bitrary integrable function ψ and it justifies use o f genera lized functions and der iv atives in Eq. (2.1). 2.2 Dirac’ s method of chords The Dirac’s metho d o f chords [14] uses tr ansition from six-dimens ional integral ov er pair of p oints in some conv ex b ody V to express io ns with chord length distribution, e.g. , D V ( ϕ ) ≡ 1 V Z V Z V ϕ ( R ) 4 π R 2 d r d r ′ = S 4 V Z ∞ 0 µ ( x )  Z x 0 Z p 0 ϕ ( r ) dr dp  dx, (2.5) where R = | r ′ − r | is dis tance b etw een p oints, d r and d r ′ are tw o three-dimensiona l volume elements, S is surface are a , and V is volume of V . Such an equatio n was derived in Ref. [14] for particular function ϕ ( R ) = exp( − αR ). It is s ho wn in Appe ndix A, that der iv ation o f Eq. (2.5) has direc t connection with equalities Eq. (2.1). Here is only outlined bas ic results and few s teps o f deriv ation Eq. (2.5). First, the integral D V may b e e xpressed via function of distances η ( l ). F rom Eq. (A2) with Eq. (A4) follows quite understanding rela tion 1 V D V ( ϕ ) = 1 V 2 Z V Z V ϕ ( R ) 4 π R 2 d r d r ′ = Z ∞ 0 ϕ ( x ) 4 π x 2 η ( x ) dx. (2.6) Due to Eq. (A7) with Eq . (A4) rig ht-hand side of Eq. (2.6) may b e rewritten using a utocor re- lation function γ ( l ) E q. (A5) D V ( ϕ ) = Z ∞ 0 γ ( x ) ϕ ( x ) dx (2.7) c  A. ⁀ Yu.Vlasov, 200 7 4 Si gned CLD The six-dimensio na l integral D V may b e reduced to a fo ur -dimensional one by tw o s teps [1 4] revisited in de ta ils in Appendix A. In accor dance with L emma 1 in Appendix A-1, Eq. (A9) a nd Eq. (A4) it may b e expres sed D V ( ϕ ) = 1 4 π V Z V d r Z d Ω Z R max 0 ϕ ( R ) dR, (2.8) where R max is length of r a dius for given po in t and direction, d r d Ω is five-dimensional integration on a ll p oints and dire c tions. Next, L emma 2 in Appendix A-3 let us rewrite Eq. (2.8 ) using ra dii density function ι ( l ) D V ( ϕ ) = Z ∞ 0 ι ( x )  Z x 0 ϕ ( r ) dr  dx. (2.9) Finally , due to rather standard arguments [14], r evisited in Appendix A-5, it is p ossible to rewrite Eq. (2.8) using four -dimensional in tegr ation on space of lines T D V ( ϕ ) = 1 4 π V Z d T Z L ch 0 dp Z p 0 ϕ ( r ) dr, (2.10) where d T is canonical inv ariant measure on the space of lines a nd L ch is length o f chord for g iven line intersecting b o dy V . Now it is p ossible to r ewrite Eq. (2.10) using L emma 5 from App endix A-5 and E q . (A33) to pro duce initial Eq. (2.5). These integrals also justify use genera lized functions, b ecause ma y b e asso ciated with lin- ear functionals on some test function ϕ . The rela tions b et ween integrals may be consider e d as transformatio ns of these functionals without necessity to indicate a n y par ticular ϕ and it is q uite reasona ble, becaus e main purp ose of this pap er is rather discussion ab out geometr ic al distributions than ab out ca lculation o f some integrals. The Eq. (2.7) may b e rewritten using Eq. (2.3) for spa ce of test functions defined on some int erv al 0 ≤ x ≤ l max D V = T γ ≡ γ . (2.11a) The Eq. (2.9) may b e rewr itten due to definition o f generalize d deriv ative Eq. (2.4) D V ( φ ′ ) = Z ∞ 0 ι ( x ) φ ( x ) dx = ⇒ D ′ V = − T ι ≡ − ι. (2.11b) Finally , from Eq. (2.5) follows D V ( φ ′′ ) = S 4 V Z ∞ 0 µ ( x ) φ ( x ) d x = ⇒ D ′′ V = S 4 V T µ ≡ S 4 V µ = 1 h l i µ. (2.11c) The Eq. (2.11) co rresp ond to Eq. (2.1) for gener alized functions and deriv a tives. The der iv ation of integral Eq. (2.7) with auto correlation function γ ( l ) is quite str a igh tforward and revisited in App endix A-2 Eq. (A7). The Dirac expr ession E q . (2.5) is directly derived fr o m Eq. (2.7) via t wo integration by parts if Eq. (2.1) is true. So if we could consider Eq. (2.1) as a “definition” of a function µ ( l ) via seco nd deriv atives of γ ( l ) it ensur es Eq. (2.5). Such a prop erty was use d in some works for definition of for mal (generaliz e d) chord length distribution via E q . (2.1) for b o dy with arbitrar y shap e and density [5, 6]. c  A. ⁀ Yu.Vlasov, 200 7 5 Si gned CLD 3 NONCONVEX BOD Y 3.1 Formal integration by parts Let us consider application o f Eq. (2.5) to some nonc o n vex b o dy N . The distance s distribution and auto correlation function is defined for nonc o n vex b o dy a nd so it is p ossible to write D N ( ϕ ) ≡ Z N Z N ϕ ( R ) 4 π V R 2 d r d r ′ = V Z ∞ 0 ϕ ( x ) 4 π x 2 η ( x ) dx (3.1a) = Z ∞ 0 γ ( x ) ϕ ( x ) dx (3.1b) = − Z ∞ 0 γ ′ ( x )  Z x 0 ϕ ( r ) dr  dx (3.1c) = Z ∞ 0 γ ′′ ( x )  Z x 0 Z p 0 ϕ ( r ) dr dp  dx. (3.1d) Here Eq. (3.1a) and Eq. (3.1b) coincide with Eq. (2.6) and Eq. (2.7) in convex case re spectively , but Eq. (3.1c) and Eq. (3.1d) ar e pro duced by formal integrations by pa rts and need for geometrica l int erpr etation. Let us deno te for nonco nvex b ody the signed r a dii a nd chord densities repr e s en ted via (normalized) first and second deriv ative as ι ± ( l ) and µ ± ( l ) resp ectively . 3.2 Radii (signed) density function An analog ue of Eq . (2.8) used for int ro duction of radii density function has form D N ( ϕ ) = 1 4 π V Z N d r Z d Ω Z R ∩ N ϕ ( R ) dR, (3.2) where R ∩ N is int er s ection of a bo dy N with a ray from a p oint inside the b o dy Fig. 3. R 1 R 2 R 3 Figure 3: Scheme of interv als for r a dii in nonco n vex bo dy It is p ossible to write Z R ∩ N ϕ ( R ) dR ≡ n I X j =0 Z R 2 j +1 R 2 j ϕ ( R ) dR, R 0 ≡ 0 , (3.3) where n I is amount of interv als [ R 2 j , R 2 j +1 ] of ray inside b ody N with R k for k > 0 cor respo nding to 2 n I − 1 p oin ts of intersection of ray with s ur face of b ody Fig. 3. The Eq. (3.3) may b e forma lly rewritten n I X j =0 Z R 2 j +1 R 2 j ϕ ( R ) dR = 2 n I − 1 X k =1 ( − 1) k − 1 Z R k 0 ϕ ( R ) dR, (3.4) c  A. ⁀ Yu.Vlasov, 200 7 6 Si gned CLD Let us denote n max maximal p ossible amount of interv als for giv en b ody and define ι k ( l ), k = 1 , . . . , 2 n max − 1 as density function for k -th interv al length R k . Then it is p ossible to der iv e from Eq. (3.2) for ι ± ( l ) = 2 n max − 1 X k =1 ( − 1) k − 1 ι k ( l ) (3.5) analogue of Eq . (2.9) D N ( ϕ ) = 2 n max − 1 X k =1 ( − 1) k − 1 Z ∞ 0 ι k ( x )  Z x 0 ϕ ( r ) dr  dx = Z ∞ 0 ι ± ( x )  Z x 0 ϕ ( r ) dr  dx (3.6) It is conv enient to consider three distr ibutions ι 1 ( l ), ι + ( l ), and ι − ( l ), where ι − ( l ) = n max − 1 X k =1 ι 2 k ( l ) , ι + ( l ) = n max − 1 X k =1 ι 2 k +1 ( l ) , ι ± ( l ) = ι 1 ( l ) + ι + ( l ) − ι − ( l ) . (3.7) Here “p ositive” and “nega tiv e” distributions, i.e. , ι 1 ( l ) + ι + ( l ) and ι − ( l ) res p ectively may b e nonzero for the sa me l , e.g. , some radii R 2 may b e equal to R 3 in o ther p oints. So Eq. (3.7) could not be consider ed as true J ordan decomp osition defined as difference of tw o no nnegative functions with nonoverlapping supp ort [12] ι ± ( l ) = ι + ( l ) − ι − ( l ) , ∀ l : ι + ( l ) ≥ 0 , ι − ( l ) ≥ 0 , ι + ( l ) ι − ( l ) = 0 . (3.8) In fact, for so me nonconv ex b o dies ι ± ( l ) ≥ 0 and so ι − ( l ) ≡ 0 despite o f no nze ro ι − ( l ) b ecause of ι 1 ( l ) + ι + ( l ) ≥ ι − ( l ). An yway , the scheme discussed ab ov e and expressed by Fig. 3, Eq. (3.5), Eq. (3.6), and Eq . (3.7) provides a sto c hastic int er pretation of ι ± ( l ). Similar with Definition 2 there is a n uniform distri- bution of p oints ins ide a b ody and is otropic rays, co nsidered a s some kind of “primar y even ts” for radius R 1 and ι 1 ( l ). E ac h r a y intersecting b ody N mor e than one time n I > 1 als o pro duces tw o kinds o f “ secondary” even ts: n I − 1 radii R 2 k +1 from a “ p ositive” distributio n ι + ( l ) and n I − 1 radii R 2 k from a “negative” one ι − ( l ). Such a sto c hastic mo del also pro duces understanding description of s ome in tegr als via averages, mathematical exp ectations etc ., e.g. , Z ∞ 0 ι 1 ( l ) dl = Z ∞ 0 ι ± ( l ) dl = 1 , Z ∞ 0 ι + ( l ) dl = Z ∞ 0 ι − ( l ) dl , (3.9) bec ause ea ch r adius corr esponds to n I “p ositiv e” even ts and n I − 1 “negative” even ts, i.e. , contri- bution to “total charge” or “ balance” of even ts N ≡ N + − N − is alwa ys o ne. It is a lso us eful to introduce distribution of total length of all segments for given radii ι O ( l ) (OSD, one-seg men t distribution), then h l i ι ≡ Z ∞ 0 ι ± ( l ) l dl = Z ∞ 0 ι O ( l ) l dl , (3.10) bec ause contribution for a n y ray is P 2 n I − 1 k =1 ( − 1) k − 1 R k = R 1 + P n I − 1 k =1 ( R 2 k +1 − R 2 k ), i.e. , total length of all se gmen ts. c  A. ⁀ Yu.Vlasov, 200 7 7 Si gned CLD 3.3 Chor d leng th (signed) density function An analogue of in tegral Eq. (2.10) for nonconv ex b ody N may not use contin uous area of in tegra tio n on dp dr , if chord int er s ects N a long few interv a ls. It is similar with in tegr ation along s et of interv al for radii Eq. (3.2) and Eq. (3.3) in Sec. 3.2, but decomp osition is more co mplex b ecause of tw o int egr als. F or a chord with n I int erv als there ar e 2 n I po in ts of intersection ( L 0 ≡ 0 , L 1 , . . . , L 2 k , L 2 k +1 , . . . ) for k = 0 , . . . , n I − 1. In Eq. (2.10) p is co ordina te along the chord and r is distance b et ween po in ts. If to intro duce new v ar iables x = p , x ′ = p + r , then for whole chord inside b o dy N L ( ϕ ) ≡ Z L 0 Z p 0 ϕ ( r ) dr dp = − Z L 0 Z x ′ 0 ϕ ( x ′ − x ) dx dx ′ = − Z L 0 Z L x ϕ ( x ′ − x ) dx ′ dx, (3.11) but for few s e gmen ts the integration of ϕ ( r ) = ϕ ( x ′ − x ) on dx dx ′ should include all p oin ts x, x ′ ∈ L ∩ N , x < x ′ , where L ∩ N = S n I − 1 k =0 [ L 2 k , L 2 k +1 ] is unio n of all int er v als o f given chord inside the b o dy , Fig. 4a. Here the minus signs in E q. (3.11) a r e due to ter m − x in ϕ ( x ′ − x ). a) b) x L 0 L 1 L 2 L 3 L 4 L 5 x c) Figure 4: a ) Scheme of in tegr ation. b) Scheme of interv als c) Decomp osition of  j,k Let us denote ( L ∩ N ) 2 ⋌ area of integration des cribed ab ov e a nd depicted o n Fig. 4a as set of triangles and recta ngles b elo w of the diagonal x = x ′ ( L ∩ N ) 2 ⋌ = n I − 1 [ k =0 N k S k l and unit otherwis e. F or such a function I V int egr ates over all pa irs of p oints in V × V with dista nc e s less than l and due to Eq . (A2) it is po ssible to write equa tio n for pro babilit y I V (Θ ¬ l ) = P ( | r ′ − r | < l ) = R l 0 η ( x ) dx ≡ F η ( l ) and it coincides with definition o f density η ( x ) for distances distribution function F η ( x ). F or extensio n of the proof for non-regular ca se, it is possible to use directly the distances distribution function F η ( x ) a nd to wr ite instea d of E q . (A2) Leb esgue-Stieltjes integral [1 2, 23] I V (Φ) = Z ∞ 0 Φ( x ) dF η ( x ) . (A3) F or consideration of both regular a nd non-re g ular ca ses it is also possible to treat η as a generalized function. In such a c a se representation of I V (Φ) as an integral Eq . (A2) may lo ok as a not very rigor representation o f a tautology like I V = η ≡ T η , b ecause a ge ne r alized function by definition is a linea r functional [12]. The functional I V is connec ted with D V defined in Sec. 2.2, Eq. (2.5) by straightforward relations D V ( ϕ ) = V I V  ϕ ( x ) 4 π x 2  , I V (Φ) = 1 V D V  4 π x 2 Φ( x )  , (A4) pro duced by the choise Φ( x ) = V ϕ ( x ) 4 π x 2 . The meaningfulness L emma 1 is not o nly integral represe ntation E q. (A2), but a ssoc ia tion of I V with distribution of distances . It is s imple demo ns tration of so me metho ds to av oid B ertrand-like paradoxes, b ecause η ( x ) is not o nly p ossible dens it y for distribution o f dista nc e s b et ween p oints in a bo dy . c  A. ⁀ Yu.Vlasov, 200 7 14 Signed CLD F or calculation of I V Eq. (A1) is used integration on six-dimensiona l spa ce with na tural Eu- clidean measure on R 6 = R 3 × R 3 and it is in a greement with indep endent uniform distributions of b oth p oints used in Definition 1 . An example of alterna tive measure was used in Ref. [16]: it was considered distribution of chords like in Defi nition 3 ab ov e and segment of line b etw ee n pair of points on the chords. A distribution function for lengths of the segments, i.e. , distances b et ween e nding p oint s may b e also calculated using Heavyside s tep function [1 6], but density is not equiv a len t with η ( x ). Y et another ex ample of similar mea sure may b e constructed if we co nsider first po in t r from uniform dis tr ibution, but the second one r ′ is g enerated with is o tropic distribution of relative directions R ≡ r ′ − r a nd unifor m dis tribution of distances | R | ≡ | r ′ − r | b et ween p o in ts. In such a case distributions of p oints r and r ′ are corr elated and relatively to natura l E uclidean meas ure on ( r , r ′ ) ∈ R 3 × R 3 = R 6 ⊃ V × V here is necessar y to introduce multiplier | R | − 2 = | r ′ − r | − 2 . Up to co nstan t normalizing multiplier the alter nativ e densit y of distances intro duced ab ov e may b e expressed a s η ( x ) /x 2 , i.e. , pro p ortiona l to auto correla tion function [1, 4 , 5, 6] discussed further. A-2 Autocorrelation function F or b ody with density ρ ( r ), r ∈ R 3 is defined the auto correlatio n function γ ( r ), r ∈ R 3 or γ ( l ), l ∈ R γ ( r ) = Z R 3 ρ ( r ′ ) ρ ( r + r ′ ) d r ′ , γ ( l ) = 1 4 π l 2 Z S l γ ( r ) d Ω , d Ω = s in θ dθ dφ, (A5) i.e. , γ ( l ) is an av era ge o f γ ( r ) on sphere with r adius l , { S l : | r | = l } . In simplest case of co nstan t density ρ ( r ) = 1 for r ∈ V and zero otherwis e . F or such a case γ (0) = V , and it is co n venien t to resca le  ( r ) ≡ ρ ( r ) / √ V for normaliza tion γ (0) = 1. It is po ssible to rewr ite Eq. (A1) I V (Φ) = 1 V Z R 3 Z R 3  ( r )  ( r ′ )Φ  | r ′ − r |  d r d r ′ = 1 V Z R 3 Z R 3  ( r )  ( r + R )Φ  | R |  d r d R ( R = r ′ − r ) = 1 V Z R 3 γ ( R )Φ  | R |  d R (A6) = 4 π V Z ∞ 0 l 2 γ ( l )Φ( l ) dl, (A7) where Eq. (A7) is pro duced fro m Eq . (A6) by in tegr ation ov er spheres S l . Compariso n of Eq. (A7) and Eq. (A2) with arbitra ry function Φ( l ) pro duces r e lation b et ween γ ( l ) a nd η ( l ) η ( l ) = 4 π V l 2 γ ( l ) (A8) already mentioned ea r lier in Sec. 2.1. c  A. ⁀ Yu.Vlasov, 200 7 15 Signed CLD A-3 Distribution of radii Let us introduce new v ar iable R = r ′ − r in E q. (A1) and rewr ite integral o n d R using spherica l co ordinates I V (Φ) = 1 V 2 Z V d r Z π 0 sin θ dθ Z 2 π 0 dφ Z R ( r , θ,φ ) 0 R 2 Φ( R ) dR = 1 V 2 Z V d r Z S d Ω Z R ( r , Ω ) 0 R 2 Φ( R ) dR, (A9) where R 2 sin θ is J acobian in spherical co ordinates ( R , θ , φ ) of vector r ′ − r , S is unit sphere, and R ( r , θ , φ ) = R ( r , Ω ) is leng th of ra dius (seg men t) mentioned in Definition 2 , i.e. , distance from po in t r to surface of b o dy in direction repr esen ted by spheric a l angles θ , φ or unit vector Ω ∈ S , Fig. 6. r R Figure 6: Scheme of integration along radii with length R ( r , θ , φ ) = | R | Let us introduce linear functiona l I ( I ) V (Ψ) = 1 4 π V Z V Z S Ψ[ R ( r , Ω )] d r d Ω , (A10) where r ∈ V ⊂ R 3 , Ω ∈ S , and R ( r , Ω ) is no tation for leng th of r a dius introduced in Eq. (A9). Here I ( I ) V (1) = 1 and 4 π V is norma liz a tion, i.e. , 5D volume o f set V × S with resp ect to canonical measure on R 3 × S . It is analog ue o f norma lization of E q . (A1) with V 2 , i.e. , 6D volume o f set V × V with r espect to ca nonical mea sure on R 3 × R 3 . Lemma 2. Th e line ar functional I ( I ) V (Φ) define d by Eq. (A10) may b e r ewritten I ( I ) V (Ψ) = Z ∞ 0 Ψ( x ) ι ( x ) dx (A11) wher e ι ( x ) is density of r adii distribution in b o dy V intro duc e d in Definition 2. Pr o of. The pro of is simila r with L emma 1 . The Eq. (A11) is a gain almost tauto lo gy for generalized functions I ( I ) V = ι ≡ T ι . Let us anew co nsider I ( I ) V with Heavyside step function Θ ¬ l ( x ), to integrate ov er p oints in V × S as socia ted with radii less than given l . The measur es of integration in I ( I ) V corres p ond to uniform distribution of p oint s r a nd isotro pic distribution o f direc tio ns Ω on unit s pheres (due to transition to spherical co ordinates in s econd int egr al). It is in agr eemen t with Definition 2 and so I ( I ) V (Θ ¬ l ) = P ( R ( r , Ω ) < l ) = R l 0 ι ( x ) dx ≡ F ι ( l ) and meets definition of density ι ( x ) for ra dii leng th distribution function F ι ( x ). c  A. ⁀ Yu.Vlasov, 200 7 16 Signed CLD The functiona l I ( I ) V Eq. (A10) let us simplify Eq. (A9) I V (Φ) = I ( I ) V  4 π V Z x 0 R 2 Φ( R ) dR  (A12) and substition of E q. (A12) to E q. (A11) pro duces yet ano ther expressio n fo r I V (Φ) I V (Φ) = Z ∞ 0  4 π V Z x 0 R 2 Φ( R ) dR  ι ( x ) dx. (A13) A-4 Autocorrelation function and distribution of r adii Due to definition E q. (A8) it is p ossible to r ewrite E q. (A2) I V (Φ) = Z ∞ 0 Φ( x ) 4 π V x 2 γ ( x ) dx. (A14 ) Y et ano ther expressio n I V (Φ) = − Z ∞ 0  4 π V Z x 0 R 2 Φ( R ) dR  γ ′ ( x ) dx, (A15) may b e pro duced from Eq. (A14) using integration by parts. The equality of Eq. (A15) and Eq. (A13) cor respo nds to s econd equa tion in Eq . (2.1) − ι ( x ) = γ ′ ( x ) . (A16) The in teg ral repr e sen tations a b ov e let us also consider Eq. (A16 ) as deriv ative of generalized function. A-5 Chord length distrib ution Let us change or der of integration in Eq. (A9) I V (Φ) = 1 V 2 Z S d Ω Z V d r Z R ( r , Ω ) 0 R 2 Φ( R ) dR (A17) and for a ny unit vector Ω ∈ S to decomp ose spa tia l integral R d r = R R dP dn o n t wo integrals: along the axis n = n Ω parallel to Ω and on the plane P = P Ω per pendicular to Ω Fig. 7. Ω P n Figure 7: Lines a nd chords c  A. ⁀ Yu.Vlasov, 200 7 17 Signed CLD After such decomp osition Eq. (A17) may b e rewritten a s I V (Φ) = 1 V 2 Z S d Ω Z P Ω ( V ) dP Z L ( p Ω , Ω ) 0 dn Z n 0 R 2 Φ( R ) dR, (A18) where P Ω ( V ) is pro jection of V on plane P Ω and L ( p Ω , Ω ) is leng th of chord o f line defined b y direction Ω ∈ S and p oint p Ω ∈ P Ω , see Fig. 7. It is p ossible to define four-dimensio na l s pace T of (directed) lines [8, 9, 10, 24, 25, 26], there each line is defined by p oint on s phere Ω ∈ S a nd p oint of intersection p Ω with orthogo nal plane P Ω ⊥ Ω . Here each line is represented twice with tw o o pposite directions ± Ω . It is p ossible to int ro duce spa ce of undirected lines as a quotient space ˘ T = T / { +1 , − 1 } , but in mo s t expr essions below for conv enience of calcula tio ns is used T . The first tw o integrals in Eq. (A18) corr espond to int egr ation on this space I V (Φ) = 1 V 2 Z T ( V ) d T Z L ( l T ) 0 dL Z L 0 R 2 Φ( R ) dR, (A19) where T ( V ) is set o f lines int er s ecting V , L ( l T ) is c hord length for a line l T ∈ T , and d T is ca nonical (uniform and isotropic) measure o n T uniq ue defined up to constant multiplier by inv ariance with resp ect to translations and rotatio ns [8, 9, 10, 2 6]. Let us introduce linear functiona l I ( II ) V (Υ) = 1 V [ T ( V )] Z T ( V ) Υ[ L ( l T )] d T , V [ T ( V )] = Z T ( V ) d T , (A20) where V [ T ( V )] is normaliza tion, i.e. , 4D volume of T ( V ) with r espect to d T . It may b e wr itten also due to definition o f R T ( V ) d T ab ov e V [ T ( V )] = Z S d Ω Z P Ω ( V ) dP = 4 π h S P Ω ( V ) i (A21) where h S P Ω ( V ) i is average surface of pro jection of b o dy V . F or conv ex b o dies due to a Cauch y formula [8, 9] h S P Ω ( V ) i = 1 4 S, (A22) where S is sur face ar ea of V and s o V [ T ( V )] = π S. (A23) Lemma 3. Th e line ar functional I ( II ) V (Υ) define d by Eq. (A20) may b e r ewritten I ( II ) V (Υ) = Z ∞ 0 Υ( x ) µ ( x ) dx (A24) wher e µ ( x ) is density of chor d length distribution in b o dy V intro duc e d in Definition 3. Pr o of. Similar with L emma 1 a nd L emma 2 the Eq . (A24) may b e repres en ted in rather trivial form I ( II ) V = µ ≡ T µ for generalize d functions. Here again the Heavyside step function Θ ¬ L ( x ) selects subset in T ( V ) asso ciated with chords shorter than g iven L . The mea sure of integration d T in I ( II ) V corres p onds to uniform and isotro pic distr ibution of lines l T in ag reemen t with D efinition 3 and so I ( II ) V (Θ ¬ L ) = P ( L ( l T ) < L ) = R L 0 µ ( x ) dx ≡ F µ ( l ) a nd meets definition of density µ ( x ) for chord length distribution function F µ ( x ). c  A. ⁀ Yu.Vlasov, 200 7 18 Signed CLD The space of lines has nont r iv ial structure and an essential mo men t in pr oof of L emma 3 is corres p ondence of meas ure of int eg r ation d T in E q. (A20) and mea sure describing distribution of lines in D efinition 3 . This iso tropic uniform distribution sometimes asso ciated with concept of µ -randomness [1, 6, 15, 1 9]. An alternative distribution of lines is o ften defined by uniform distribution o f p oin ts ins ide the bo dy V and isotropic distr ibution of dire c tio ns. It is so metimes called ν -randomness [1, 19] o r int erio r ra diator randomness [1 5]. It is different from µ - randomness, b ecause the same line may be re pr esen ted by any p oint o n this line and direction alo ng this line. So measure of ν -c hor d for given line in compar ison with unifor m case has ex tra multiplier pro p ortiona l to leng th of the chord ν ( l ) ∝ l µ ( l ). The pre c ise expres sion for conv ex b ody is [1, 15, 19] ν ( l ) = l h l i µ ( l ) =  4 V S  − 1 l µ ( l ) . (A25) The co nce pt o f ν -randomness for chords is clo s e with Defin ition 2 of ra dii distribution and it is useful for some applica tions [15, 1 9]. Y et another distribution is pro duced by definition of line by pair of p o in ts with indepe nden t uniform distributions inside V . It is so metimes calle d λ -ra ndomness [1, 6, 19]. The measure for λ -chord for three-dimensio nal ca se λ ( l ) ∝ l 4 µ ( l ). The nor malizing multiplier for l 4 may b e dir e c tly calculated [1, 3, 1 0, 14, 19] h l 4 i = 12 V 2 π S (A26) and precise express ion for conv ex b ody is [1, 19] λ ( l ) = l 4 h l 4 i µ ( l ) =  12 V 2 π S  − 1 l 4 µ ( l ) . (A27) The concept o f λ -ra ndomness for chords has certain r elation with D efinition 1 of distance s distri- bution. Such different kinds of rando mness illustra tes nece s sit y of rather pedantic work with distribution of lines due to ana lo gues of Ber trand para dox [8, 1 3] already mentioned earlier . Lemma 4. Th e functional Eq. (A20) may b e formal ly expr esse d via Eq. (A10) I ( II ) V  Z x 0 Ψ( l ) dl  = 4 V S I ( I ) V (Ψ) . (A28) Pr o of. The deriv atio n of E q . (A28) is similar with tra nsition from E q . (A9) to E q . (A19) via Eq. (A17) and Eq. (A18 ) 4 V S I ( I ) V (Ψ) = 4 V S 1 4 π V Z V Z S Ψ[ R ( r , Ω )] d r d Ω = 1 π S Z S d Ω Z V d r Ψ[ R ( r , Ω )] = 1 π S Z S d Ω Z P Ω ( V ) dP Z L ( p Ω , Ω ) 0 dn Z n 0 Ψ( l ) dl = I ( II ) V  Z x 0 Ψ( l ) dl  . An application Eq. (A24) and Eq. (A11) to Eq . (A28) pro duces Z ∞ 0  Z x 0 Ψ( l ) dl  µ ( x ) dx = 4 V S Z ∞ 0 Ψ( x ) ι ( x ) dx. (A29) c  A. ⁀ Yu.Vlasov, 200 7 19 Signed CLD An integration by pa rts of Eq. (A29) pro duces Z ∞ 0  Z x 0 Ψ( l ) dl  µ ( x ) dx = − 4 V S Z ∞ 0  Z x 0 Ψ( l ) dl  ι ′ ( x ) dx. (A30) The first equality in Eq. (2.1) follows fro m Eq. (A30) ι ′ ( x ) = −  4 V S  − 1 µ ( x ) = − 1 h l i µ ( x ) . (A31) It may b e considered as genera lized deriv ative due to E q. (A29). In E q. (A31) w as used Ca uc hy relation Eq. (2.2) 4 V /S = h l i . It is p ossible also to find normalizing multiplier n µ for n µ µ ( x ) = − ι ′ ( x ) simply using integration by par ts n − 1 µ = n − 1 µ Z ∞ 0 ι ( l ) dl = − n − 1 µ Z ∞ 0 ι ′ ( l ) l dl = Z ∞ 0 l µ ( l ) dl = h l i . (A32) This deriv atio n uses only the fact of propo rtionality of µ ( x ) to deriv ative o f ano ther dens it y function and in further applications for nonco n vex ca se normaliza tion with h l i may b e preferable due to nontrivial pro of of expressio n with volume and sur face area. Lemma 5. Th e functional I V Eq. (A1) may b e expr esse d via I ( II ) V Eq. (A 20) I V (Φ) = S 4 V I ( II ) V Z x 0  Z l 0 4 π V r 2 Φ( r ) dr  dl ! . (A33) Pr o of. The expressio n E q. (A33) is straightforward combination of E q . (A12) a nd Eq. (A28). Application of Eq. (A24) to E q. (A33) pro duces I V (Φ) = S 4 V Z ∞ 0 Z x 0 Z l 0 4 π V r 2 Φ( r ) dr dl ! µ ( x ) dx. (A34) The E q. (A1) a nd Eq . (A34) pro duce for Φ( r ) = V 2 ϕ ( r ) 4 π r 2 Z V Z V ϕ  | r ′ − r |  4 π | r ′ − r | 2 d r d r ′ = S 4 Z ∞ 0 µ ( x )  Z x 0 Z p 0 ϕ ( r ) dr dp  dx, (A35) in agre emen t with E q . (2.5) in Sec. 2.2. It is a lso conv enient sometimes to use expressio n Z x 0 Z p 0 ϕ ( r ) dr dp = Z x 0 ( x − r ) ϕ ( r ) dr . (A36) APPENDIX B SOME EQU A TIONS FOR NONUNIFORM CASE Let us c o nsider a b o dy with nonuniform density ρ ( r ). Any such b o dy N may b e treated as a conv ex one without lost o f genera lit y by consideration o f the co n vex hull V and assignment o f zero density to the co mplemen t V \ N . Here is considered tw o cases corr esponding to choice of ∆ r , r ′ in Eq. (4.1). c  A. ⁀ Yu.Vlasov, 200 7 20 Signed CLD B-1 Distance between points ∆ r ,r ′ = | r ′ − r | A direct analog ue o f E q. (A7) is Z Z ρ ( r ) ρ ( r ′ ) ϕ  | r ′ − r |  4 π | r ′ − r | 2 d r d r ′ = Z ∞ 0 γ ( x ) ϕ ( x ) dx = Z ∞ 0 γ ′′ ( x )  Z x 0 Z p 0 ϕ ( r ) dr dp  dx, (B1) there seco nd equality pr oduced by tw o in tegr ations by parts. So up to normaliz a tion with R ∞ 0 γ ′′ ( x ) dx = γ ′ (0) it is p ossible to use a forma l (“generalized” [5 , 6]) chord length distribu- tion ´ µ ( l ) = γ ′′ ( l ) / γ ′ (0) for calcula tion o f integrals like Eq. (B1). Finally Z Z ρ ( r ) ρ ( r ′ ) ϕ  | r ′ − r |  4 π | r ′ − r | 2 d r d r ′ = ´ C µ Z ∞ 0 ´ µ ( x )  Z x 0 Z p 0 ϕ ( r ) dr dp  dx, ´ C µ = γ ′ (0) . (B2 ) There is yet another way to express ´ C µ . If to consider ϕ ( l ) = 4 π l 2 , then from E q . (B2) follows M 2 = Z Z ρ ( r ) ρ ( r ′ ) d r d r ′ = ´ C µ Z ∞ 0 ´ µ ( x )  Z x 0 Z p 0 4 π r 2 drdp  dx = π 3 ´ C µ Z ∞ 0 x 4 ´ µ ( x ) dx. where M ≡ R ρ ( r ) d r is mass of the b o dy . Let us als o denote ´ h l 4 i ≡ R ∞ 0 x 4 ´ µ ( x ) dx , then ´ C µ = 3 M 2 π ´ h l 4 i . (B3) F or case of cons tan t unit density M = V and due to Eq. (A26) h l 4 i = 12 V 2 / ( π S ). So C µ = S/ 4 in agre emen t with E q . (A35) a nd Cauchy formula Eq. (A22). B-2 “Optical” length ∆ r ,r ′ = O r ′ r In some applications instea d o f distance b et ween po in ts R = | r ′ − r | in ϕ ( R ) it is necessary to use an “optical width”, i.e. , integral on density alo ng line b etw een p oin ts O r ′ r = r ′ ∫ r ρ dℓ = | r ′ − r | Z 1 0 ρ  r + ( r ′ − r ) x  dx (B4) and to consider functiona l J ( ϕ ) = Z Z ρ ( r ) ρ ( r ′ ) ϕ ( O r ′ r ) 4 π | r ′ − r | 2 d r d r ′ (B5) Let us introduce new v ar iable and rewr ite integral o n d R using spher ical co ordinates like in Eq. (A9) J ( ϕ ) = 1 4 π Z R 3 d r Z S d Ω ρ ( r ) Z R ( r , Ω ) 0 ρ ( r + x Ω ) ϕ ( O r + x Ω r ) dx. (B6) where R ( r , Ω ) is lenght of (ma ximal) radius, O r + x Ω r = R x 0 ρ ( r + l Ω ) dl ≡ s ( x ), s ′ ( x ) = ρ ( r + x Ω ). Last integral has for m R l 0 s ′ ( x ) ρ ( s ( x )) dx and may b e rewr itten as R s ( l ) 0 ρ ( s ) ds . J ( ϕ ) = 1 4 π Z R 3 d r Z S d Ω ρ ( r ) Z w ( r , Ω ) 0 ϕ ( w ) dw , (B7) c  A. ⁀ Yu.Vlasov, 200 7 21 Signed CLD where w ( r , Ω ) = O r + R ( r , Ω ) Ω r = R R ( r , Ω ) 0 ρ ( r + l Ω ) dl is “ optical” ra dius length. It is p ossible to use analog ue of integration in E q. (A18) with axis n k Ω and plane P ⊥ Ω J ( ϕ ) = 1 4 π Z S d Ω Z P dP Z dn ρ ( r ) Z w ( r , Ω ) 0 ϕ ( w ) dw , r = r P + n Ω , (B8) where p oint r P is intersection (pro jection) of line repres e n ting path of integration with plane P (see Fig. 7) a nd ra nge of n corres p onds to v ariation of r along whole chord in conv ex span o f a bo dy . If to conside r last integral in E q. (B8) as some function F ( w ( n )), tw o la s t integrals hav e form R w ′ ( n ) F ( w ( n )) dn = R F ( w ) dw a nd so we hav e ana lo gue of Eq. (A19) J ( ϕ ) = 1 4 π Z T d T Z W ( l T ) 0 dw Z x 0 ϕ ( x ) dx, (B9 ) where W ( l T ) = O r max r min ( l T ) is “optical” chord leng th. Lemma 6. L et us now intr o duc e “optic al” chor d length distribution ˜ µ ( x ) , i.e., to any chor d of line interse cting (c onvex hul l of ) a b o dy in t wo p oints r 1 and r 2 inste ad of | r 2 − r 1 | is assigne d “optic al” length O r 2 r 1 . The inte gr al Eq. (B5) may b e ex pr esse d J ( ϕ ) = ˜ C µ Z ∞ 0 ˜ µ ( x )  Z x 0 Z p 0 ϕ ( r ) dr dp  dx. (B10) Pr o of. The Eq. (B10) follows from Eq. (B9) a nd it is complete analogue of deriv a tion E q. (A34) from E q. (A19 ) in App endix A . In b oth ca ses is used the sa me integral of so me function F ( l T ) ov er spa ce of lines T with the same measur e d T and pa rticular metho d of ca lc ula tion o f F do es not matter. The µ ( l ) in Eq. (A34) is (joint) density for F ( l T ) = L ( l T ) a nd the ˜ µ ( l ) in Eq. (B1 0) is (join t) density for F ( l T ) = W ( l T ). There is unessen tial difficulty with constan t multiplier ˜ C µ , due to lac k o f simple analo gue of Cauchy formula Eq. (A22 ) for average sur face use d in normalizatio n of the integral o n T in Eq. (A23). An alter nativ e wa y of c alculation o f ˜ C µ is repre sen ted b elow. Let us introduce quantit y G ≡ Z Z ρ ( r ) ρ ( r ′ ) 4 π | r ′ − r | 2 d r d r ′ = Z ∞ 0 γ ( x ) dx (B1 1) then for Eq. (B10) with ϕ ( l ) = 1 G = ˜ C µ Z ∞ 0 ˜ µ ( x )  Z x 0 Z p 0 drdp  dx = 1 2 ˜ C µ Z ∞ 0 x 2 ˜ µ ( x ) dx = 1 2 ˜ C µ ˜ h l 2 i and ˜ C µ = 2 G ˜ h l 2 i (B12) Finally Z Z ρ ( r ) ρ ( r ′ ) ϕ ( O r ′ r ) 4 π | r ′ − r | 2 d r d r ′ = ˜ C µ Z ∞ 0 ˜ µ ( x )  Z x 0 Z p 0 ϕ ( r ) dr dp  dx, ˜ C µ = 2 G ˜ h l 2 i . (B13) c  A. ⁀ Yu.Vlasov, 200 7 22 Signed CLD APPENDIX C A VERA GE P A TH LENGTH Here is presented simple geometrica l pro of of eq ua lit y of average path length inside a b ody and av erage ch or d length in uniform iso tropic ca se. On the Fig. 8a is depicted s o me path AB C with one kink inside a b o dy . Let us also co nsider path F B D pr oduced from AB C b y central s ymmetry with re spect to p oin t B , Fig. 8b. The sum of lengths of tw o paths F B D and AB C is | AB | + | B C | + | F B | + | B D | a nd co incides with sum for t wo chords AF and C D Fig. 8c. A B C D F A B C D F A B C a) b) c) Figure 8: Illustration for average path length This metho d let us g et rid of o ne kink and after few such steps to consider av erag e length of straight lines instead of paths. The necessary condition for such a pro of — is isotr opic and uniform distribution of paths to ens ure equa l pr o babilit y (density) of AB C and F B D . Referenc es [1] W. Gille, “Chord length distributions and small-angle scattering ,” Eur. Phys. J. B 1 7 , 371 – 383 (2000). [2] A. Mazzolo , B. Ro essling e r, and W. Gille, “ Prop erties of chord length dis tr ibutions o f non- conv ex b o dies,” J . Math. Phys. 44 , 619 5–6208 (2003 ). 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