On L p -affine surface areas

Transcription

1 On L p -affine surface areas Elisabeth M. Werner Abstract Let K be a convex body in R n with centroid at 0 and B be the Euclidean unit ball in R n centered at 0. We show that lim t 0 K K t = O pk O p B, where K respectively denotes the volume of K respectively B, O p K respectively O p B is the p-affine surface area of K respectively B and {K t } t 0, {B t } t 0 are general families of convex bodies constructed from K, B satisfying certain conditions. As a corollary we get results obtained in [3, 5, 6, 31]. 1 Introduction and notation During the past two decades affine surface area, originally a basic invariant from the field of affine differential geometry, has become an important tool in convex geometry. For this to happen affine surface area had first to be extended so that it was defined on all convex bodies see e.g. [8, 14,, 4, 9]. Work on the extension problem for affine surface area lead to the solution of the uppersemicontinuity problem for affine surface area [14] which has recently had an impact in the study of affine PDE s see e.g. Trudinger and Wang [7] and Wang [8]. The flurry of activity surrounding affine surface area ultimately led to the Ludwig-Reitzner Characterization theorem [9] and various deep results in the area of e.g. combinatorics see e.g. [1], polytopal approximation see e.g. [5, 13, 5] and the theory of valuations see e.g. [10, 11]. partially supported by a grant from the National Science Foundation and from a NSF Advance Opportunity Grant Mathematics Subject Classification: 5 A 0 1

2 During the past decade it has come to be seen that the classical Brunn- Minkowski theory of convex bodies is a part of a more general L p -Brunn-Minkwoski theory see e.g. [, 19, 0, 1] and recent advances in this area have influenced the work in PDE s and ODE s of Chen, Chou, Hu, Ma, Shen and Wang [3, 4, 6] among others. Within the L p -Brunn-Minkwoski theory, L p -extensions have been found of the basic affine invariants of classical convex geometry. One of the most important affine invariants for which L p -extensions have been discovered is affine surface area. For 1 < p <, L p -affine surface area was defined by Lutwak [14, 16] for all convex bodies K in R n and extended by Hug [7] to 0 p < 1: O p K = K κ K x p < x, N K x > np 1 dµx. Here κ K x is the generalized Gaussian curvature and N K x the outer unit normal in x K, the boundary of K, and µ is the surface measure on K. The next step was taken by Meyer & Werner [] who extended the definition of L p -affine surface area to all p n, and also gave a geometric definition of L p -affine surface area in terms of L p -Santaló bodies for p n,. Schütt & Werner went on to show how L p -affine surface area, for all p [, ] except p = n has a natural definition in terms of random polytopes [5], and gave another definition in terms of the surface bodies [6]. Werner [31] provided yet another definition, for all p, in terms of L p -floatation bodies. It is easy to see that O p K is finite for all p with 0 p see [6]. This need not to be so for negative values of p [6]. Moreover, O 0 K = n vol n K and for K with K C and a.e. strictly positive Gaussian curvature O ± = n vol n K 0, where K 0 is the polar body of K see below. Note also that for all p n, O p B = vol n 1 B. As remarked above, in the L p -extension process new classes of convex bodies were discovered through which L p -affine surface area can be characterized geometrically. This new geometric characterizations have a common feature: First a specific family {K t } t 0 of convex bodies is constructed. This family is different in each of the extensions but of course related to the given convex body K. The L p -affine surface area is then obtained by using expressions involving volume differences K K t. Examples for such families K t are the L p -Santaló bodies [3], the L p -floatation bodies or weighted floating bodies [31] and the surface bodies [6] see below for the definitions. Therefore it seemed natural to ask whether there are completely general conditions on a family {K t } t 0 of convex bodies in R n that in connection with volume difference expressions will give L p -affine surface area. A positive answer to this

3 question was given in [30] in the case p = 1. We show here that this also holds in the general case. The author wants to thank Monika Ludwig for initiating this paper and for many helpful discussions. She contributed so much, she should have been a coauthor. She also wants to thank the referee who suggested many improvements to the paper. Ba, r = B n a, r is the n-dimensional Euclidean ball with radius r centered at a. We put B = B0, 1 = B n 0, 1 and r B = B0, r. By. we denote the standard Euclidean norm on R n, by <.,. > the standard inner product on R n. For two points x and y in R n [x, y] = {αx + 1 αy : 0 α 1} denotes the line segment from x to y. For a set A R n, inta is the interior of A. For a convex body K R n, K denotes the boundary of K. µ is the usual surface area measure on K. B = S n 1. For x K, N K x is the outer unit normal vector to K in x. We denote the n-dimensional volume of K by K. H + and H are the closed halfspaces determined by the hyperplane H. We will assume from now on that a convex body K in R n is positioned such that the centroid of K is at the origin. We choose the centroid to be at 0 but we could have chosen any interior point of K instead. We will then denote by K the set of convex bodies in R n with centroid at the origin. Let K be a convex body in R n. For x K let rx be the radius of the biggest Euclidean ball contained in K that touches K at x. It was shown in [4] that µ-a.e. on K rx > 0 and that for 0 α < 1 rx α dµx < 1 K It was also noted in [4] that in general α cannot be chosen to be equal to 1. Therefore, µ-a.e. on K there exists a centered ellipsoid E c x such that E c x K = {x}, N K x = N Ecxx n 1 E c x = < x, N K x > n+1 rx. Moreover there exists a hyperplane H c x such that x int H c x and E c x Hc x K Hc x. 3

4 We call E c x a contained ellipsoid. It should be noted that E c x is not unique. Likewise, the hyperplane H c x such that holds, is not unique: if f.i. H c x is such that holds, then for any hyperplane H H c x parallel to H c x and with x int H, holds as well. Let K R n and x K with unique outer unit normal vector N K x and strictly positive Gauss curvature κ K x. Then see f.i. [8, 5], for ε > 0 given, there exist centered ellipsoids E i ax and E c ax such that κ E i a xx = x E i ax, x E c ax, 3 N K x = N E i a xx = N E c a xx, 4 κ Kx 1 ε n 1, κ E c a x x = κ Kx 1 + ε n 1 5 and such that Eax i and Eax c have common axes: of length < x, Nx > once 1 1 and of length 1 ε <x,nx> respectively 1 + ε <x,nx> n 1-times. κx n 1 1 κx n 1 1 Thus E i ax = 1 ε n 1 < x, Nx > n+1 κx 1 6 Eax c = 1 + ε n 1 n+1 < x, Nx > κx 1. 7 Moreover there exists a hyperplane H a x, ε such that x int H a x, ε and such that E i ax H a x, ε K H a x, ε E c ax H a x, ε. 8 We call E i ax and E c ax the approximating ellipsoids. Again, the hyperplane H a x, ε is not unique. Similar comments as above for H c x apply. If x K is such that κ K x = 0, then it is well known that the indicatrix of Dupin at x is an elliptic cylinder see [8, 4]. Hence there is a centered ellipsoid E 0 x that touches K in x and which has at least one axis that is arbitrarily large: for all ε > 0 there exists a centered ellipsoid E 0 x with at least one axis a = aε with a > 1 ε see [4]. Moreover there exists a hyperplane H 0x, ε not unique such that E 0 x H 0 x, ε K H 0 x, ε. 9 We want to emphazise that above as well as in the following definition the hyperplanes have a mostly auxiliary function: the only information we need about K is the behavior near the boundary and the role of the hyperplanes is to express just that. 4

5 Definiton and Examples Definition Let p R, p n be fixed. For every t R, t 0, let F p t : K K, K F p t K = K p t be a map. We write in short K t = K p t. We say that i F p t is p-limiting if for all invertible linear maps T, lim t 0 T B T B t = dett n p ii F p t is C-inclusion preserving, if there exists a constant C = CK 1 such that µ-a.e. on K we have: There exists t 1 such that, whenever E c x is a contained ellipsoid such that holds for some hyperplane H c x, then for all t t 1 E c x Hc x K t Hc x. Ct iii F p t is local if it is 1-inclusion preserving for the approximating ellipsoids: Let ε > 0 be given and let x K with approximating ellipsoids Eax i and Eax c such that 8 holds for some H a x, ε. Then there exits t = t ε, x such that for all t t Eax i Ha x, ε K t Ha x, ε Eax c Ha x, ε. t t iv F p t is monotone if for every centered ellipsoid E, E t is again a centered ellipsoid homothetic to E and if the radius rt = 1 ft of the centered ball B t is such that f0 = 0, f is increasing and there is a constant d independent of t such that fct d ft, where C is the constant in ii. We would like to make some comments on the definition of the map F p t. Firstly, in the definition we restrict ourselves to considering the case B t B, K t K for all t 0. A similar construction can be given for B B t, K K t for all t 0 compare [30]. Then theorems similar to Theorems 1 and below hold. 5

6 The maps F p t are essentially determined by their behavior with respect to affine images of Euclidean balls. The conditions ii and iii state that it is enough to know the behavior locally and close to the boundary of a convex body K. Globally we need only to know how the maps F p t behave for ellipsoids as stated in i and iv. Again, we want to stress that the hyperplanes H c x, H a x, ε and H 0 x, ε play mostly an auxiliary role: through them we express what counts which is the behavior near the boundary of K. Let K and L be convex bodies in K such that L K. Then we cannot expect that L t K t. We do not even have necessarily, if E c x is the contained ellipsoid, that E c x t H c x K t Hc x. This is why we require the condition C-inclusion preserving. Its geometric meaning is that, even if inclusion relation in general is not preserved passing to the floating bodies, it is preserved -at least with a constant- for the contained ellipsoids. Consider B n = {x R n : max 1 i n x i 1} and for 0 < ε < 1 let K ε be B n with rounded vertices. We describe the boundary of K ε in the first quadrantthe description in the other quadrants is accordingly: For 0 x i 1 ε, 1 i n 1, let x n = 1. For 1 ε x i 1, 1 i n 1, let n i=1 x i 1 ε = ε. Then the Euclidean unit ball B is the contained ellipsoid of K ε at e n = 0,..., 1. For fx = f 1 x = κx 1 n+1, the surface body Bf,t see the examples below of B is the Euclidean ball 1 s n t n 1 B, where sn is a constant depending on n only. For all t > 1 n 1 εn n+1 voln 1 B, K ε f,t =, hence does not contain B f,t. For t 1 n 1 εn n+1 voln 1 B, K ε f,t is contained in the half-space {x R n : x n 1 s n t n 1 ε n 1 n+1 } and thus does not contain 1 sn t n 1 B if ε is small enough. Similarily, we cannot expect that the inclusion relations 8 and 9 pass to the floating bodies. Therefore we require it as a condition - locality - which means geometrically that the inclusion relations 8 and 9 are preserved passing to the floating bodies. Locality is necessary. To see that, consider the following example see figure: Let K be the convex body in the plane which consists of a half circle and a triangle attached to it. We consider the surface body see below K fk,t of K where the function f K = f 1,K is such that it is equal to 0 on the lines of the triangle and constant equal to 1 on the half circle. Then, since K fk,0 does not contain the triangular part of K and since 6

7 B fb,t = s n t n 1 see below while K K fk,t lim t 0 B fb,t = O 1 K O 1 B = K κ 1 n+1 vol n 1 B is clearly finite. t K fk,t is not local. Let x be in the triangular part of the boundary of K. Then there is a centered ellipsoid E 0 x such that 9 holds. But E 0 x fe,0 H0 has non-empty intersection with the triangular part of K and thus is not contained in K fk,0 H0. Examples 1. The L p -Santaló bodies [3] Let t 0 be in R and let K K. Let β > n+1. Let dy S β K, t = {x K : 1 < x, y > 1 }. 10 β t K 0 Here K 0 = {y R n : < x, y > 1 for all x K} is the polar body of K. We assume again that K is C with strictly positive Gaussian curvature everywhere. Then it follows from [3] that the family of L p -Santaló bodies {K t = S β K, t} t 0, satisfiy the Definition. Indeed, if K and L are convex bodies in K such that L K, then S β L, t S β K, t. Then ii and iii of the Definition hold, ii with C = 1. We refer to [3] for the details. 7

8 If B is the Euclidean unit ball in R n, then S β B, t is again a ball with radius 1 β rt = 1 c n,β t n+1. c n,β = n 1 B n 1 0, 1 B n+1, β n+1 is a constant depending on β and n only. For x, y > 0, Bx, y = 1 0 sx 1 1 s y 1 ds is the Beta function. Thus for the L p -Santaló bodies the function f of iv is ft = 1 β c n,β t n+1 which satisfies what is required in iv. By [3] for all linear invertible maps T : S β T K, t = T S β K, dett t. This shows that for a centered ellipsoid E, E t is again a centered ellipsoid homothetic to E. Moreover 1 n t 1 1 c n,β T B T B t β n+1 dett lim t 0 = dett lim t 0 1 n β 1 1 c n,β t n+1 for p = n. β n = dett β n+3 β n+1 = dett n p,. The surface bodies [6] Let K K and f : K R be a nonnegative, integrable function. Let M f be the measure M f = fµ on K. Let t 0. The surface body K f,t is the intersection of all the closed half-spaces H + whose defining hyperplanes H cut off a set of M f -measure less than or equal to t from K. More precisely, K f,t = H + 11 M f K H t For q, q n let the functions f q : K R be given as follows: For q = ± put κ K x f ± x = f ±,K x = < x, N K x > n and for all other values of q f q x = f q,k x = κ K x q n+q < x, N K x > nq 1 n+q. We assume also that K is C with strictly positive Gaussian curvature everywhere. Then it follows from [6] that K K t = K fq,t satisfy the Definition. 8

9 Indeed, if ρb is the Euclidean ball with radius ρ, then for all q, q n, for all x ρb: f q x = ρ nq n q n+q. Then ρb t is again a ball centered t at 0 with radius rt = ρ 1 s n n 1 is a constant that ρ n n q n+q. s n = 1 B n 1 n 1 depends on n only. Thus for the surface bodies the function f of iv is ft = s n t n 1 which satisfies what is required in iv. It follows from [6], Lemma 10 that the surface body of a centered ellipsoid E is again a centered ellipsoid homothetic to E. The C-inclusion relation also holds. We give a sketch of the proof-under somewhat simplified conditions. The proof without any restrictions goes along the same lines, working with ellipsoids instead of balls and normals N K x that are not parallel to x. The calculations are just longer and more tedious to carry out but add no further insight. Let x K such that N = N K x = x. Let r = rx be the radius of the biggest Euclidean ball contained in K that touches K in x. r can be taken as r = min 1 i n a i where a i is the length of the i-th axis of the approximating ellispoid E a = E a x. Thus also κ = κx = n 1 a n i=1 see a i e.g. [5]. Let E c = E c x be the contained ellipsoid. Then, by construction of E c, for ε > 0 given, there exists a hyperplane H such that B x 1 εrn, 1 εr H E c H Bx rn, r H K H. And moreover Then, as E i a H K H E c a H n n q 1 ε r n+q µ Bx 1 εrn, 1 εr H f q,ec µ Ec r n n q n+q µ Bx rn, r H, E c H for t small enough, E c Ct H B x 1 ε rn, 1 εr H Ct = B x rn, 1 r1 s n Ct r n n q n+q n 1 H. To simplify the calculations we assume now in addition that Ea i H 1 ε 1 ε = B x N, H. κ 1 n 1 9 κ 1 n 1

10 Then E i a t H = B x 1 ε 1 ε N, 1 s n κ 1 n 1 κ 1 n 1 t 1 εκ 1/n 1 n n q n+q which is contained in K t H as for H suitably chosen f q,e i a µ E i a Ea H i f q,k µ K K H f q,e c a µ E c a Ea H c n 1 H, and for t small enough f Ea c q,e H c a µ E c a 1 + ε f Ea i q,e H i a µ E i a. Hence it is enough to show that there exists C such that B x B x rn, r1 s n Ct 1 ε 1 ε N, 1 s n κ 1 n 1 κ 1 n 1 which holds if we choose C a max a min r n n q n+q n 1 H t 1 εκ 1/n 1 n n q n+q n n q n+q a max a min n 1 H, if q n and C a max a min n n q a n+q min a max if q n where a min = min x K min 1 i n a i x and a max = max x K max 1 i n a i x. Similarly one shows the local-property. Finally, it follows from Theorem 14 and Proposition 9 of [6] that lim t 0 T B T B t O p T B n = 1 T B T B t lim t 0 ns n n 1 = dett n p O pb n t = dett n p = 3. The L p -floatation bodies [31] Let K K and t 0. Let f : K R be an integrable function such that f > 0 m-a.e. where m is the Lebesgue measure on R n. The L p -floatation body F K, f, t is the intersection of all the closed halfspaces H + whose defining hyperplanes H cut off a set of f m-measure less than or equal to t from K. More precisely, F f,t = F K, f, t = 10 R K H f dm t H + 1

11 Let p, p n and let f p x = f p,k x = < x, N Kx > nn+1p 1 κ K x np 1, for x K. If K is such that f p is continuous on K, we extend f p to all of K such that f p is continuous on K. We assume also that K is C with µ-a.e. strictly positive Gaussian curvature. Then it follows from [31] that the family {K t = F fp,t} t 0 satisfy the Definition. This is shown similarly to the surface bodies. 3 The Theorems and their proofs Theorem 1 Let 0 p < n be fixed. Let t R, t 0, let F p n t : K K, K F p t K = K t be a p-limiting, C-inclusion preserving, monotone, local map. Then K K t lim t 0 = O pk O p B. Theorem Let p n be fixed. Let t R, t 0, let F p t : K K, K F p t K = K t, be a p-limiting, C-inclusion preserving, monotone, local map. Let K K be such that K is C and has strictly positive Gaussian curvature everywhere. Then K K t lim t 0 = O pk O p B. Remarks 1. As corollaries to Theorems 1 and we obtain results of [3, 5, 6, 31]. Some of these results were obtained under weaker assumptions.. The restriction p < n in Theorem 1 is due to 1 see the proof of n Theorem 1 below. This restriction can be removed if a modified version of 1 can be proved see the Remark after the proof of Lemma 4. For the proof of Theorems 1 and we need several Lemmas. Lemma 3 Let K and L be two convex bodies in R n such that 0 intl and L K. Then 11

12 K L = 1 n where x L = [0, x] L. K < x, Nx > 1 x L x n dµx, The proof of Lemma 3 is standard. Lemma 4 Let p n be fixed. Let t R, t 0, let F p t : K K, K F p t K = K t be a p-limiting, C-inclusion preserving, monotone, local map. Let K K. Then µ-a.e. on K there exists t 0 such that 0 < x, N K x > 1 xkt n max { γ } rx pn 1 1, B t0 where x Kt = [0, x] K t and γ is a constant. Lemma 5 Let p n be fixed. Let t R, t 0, let F p t : K K, K F p t K = K t be a p-limiting, C-inclusion preserving, monotone, local map. Let K K and x K. Then i if κ K x > 0, < x, N K x > 1 x K t n lim = t 0 κ K x p < x, N K x > np 1. ii if κ K x = 0 and p > 0 or p < n < x, N K x > 1 x K t n lim = 0. t 0 x Kt = [0, x] K t. Proof of Theorems 1 and By Lemma 3 K K t lim t 0 = lim 1 t 0 n K < x, N K x > 1 x K t n x dµx. 1

13 By Lemma 4 the functions under the integral sign are uniformly bounded in t by the function { } γ gx = max rx pn 1 1,. B t0 As assumed in Theorem 1, 0 p < n and therefore g is integrable by 1. n We apply Lebesgue s convergence theorem to interchange integration and limit. By Lemma 5 the functions under the integral are converging pointwise. For the proof of Theorem note that, under the assumptions of the theorem, the functions under the integral sign are uniformly bounded in t by the constant { } γ pn 1 max r 1 0, B t0 which is integrable for all p and where r 0 = min{rx : x K}. Proof of Lemma 4 Let x K be such that r = rx > 0 and let E c = E c x be the corresponding contained ellipsoid such that holds. By the C-inclusion property there exists C and t 1 such that for all t t 1 E c x Hc x K t Hc x. Hence for all t t 1 Ct x Ec Ct x Kt. where x Ec Ct = [0, x] E c Ct and thus n x 1 Kt 1 n x EcCt By iv E c Ct is homothetic to E c : E c Ct = a E Ct E c, hence x Ec Ct Thus n x 1 Kt 1 a E Ct n. Let T be a linear invertible map such that E c = T B. = a E Ct. 13

14 As E c = < x, N K x > n+1 n 1 rx, dett = < x, N K x > n+1 T B T B Ct = E c E c Ct = B Ct B Ct dett 1 a E Ct n rx n It follows from property i that there is a constant c 1 such that for all t t = t x E c 1 a E Ct n B Ct Hence 1 a E Ct n B B Ct c1 1 n x Kt B Ct c 1 dett n p. dett p and thus for all t min{t1, t } c 1 dett p B Ct As F t is monotone, by iv there exists t 3 such that for all t t 3 B Ct = = fctn d, ft n where d is the constant from iv. Let t 0 = min{t 1, t, t 3 }. Then by 13, 14 and 15, for all t t 0 with a new constant c 1 x K t n c rx pn 1 < x, Nx > pn+1. Now notice that there is an 0 < β 1 such that B0, β K B0, 1 β. and therefore β 3 < x, Nx > 1 β. 14

15 Thus, with a new constant γ, we get for all t t 0 < x, N K x > 1 xkt n γ rx pn Let now t t 0. Then 1 x K t n 1 1 B t0. Remark For x K let H be a hyperplane through 0 with normal Nx. H is such that x H. Let Ex be a centered ellipsoid such that x Ex and such that Ex H K. Let vx be the maximum volume of such ellipsoids. Conjecture [1] Let β <. Then we have for all K in K vx β dµx <. K If the conjecture holds true, then 13 can be replaced by dett = vx and hence the right hand side of inequality 16 can be replaced by γ n p vx p. By the conjecture this is integrable, if <, which holds for all p > n. p Proof of Lemma 5 i Let x K be such that κx = κ K x > 0. Let ε > 0 be given. Let Ea i = Eax i and Ea c = Eax c be the corresponding approximating ellipsoids with properties 3-8. By the locality property iii there exists t 1 such that for all t t 1 E i ax t H a x K t H a x E c ax Hence for all t t 1 as in the proof of Lemma 4 n n x 1 E c a t x 1 Kt x 1 E i a t 15 t H a x n

16 By iv Ea i t is homothetic to Ea i and Ea c t is homothetic to Ea: c Ea i t = a i t Ea i and Ea c t = a c t Ea c and hence x Ea i t = a i t and x Ea c t = a c t. Thus n 1 a c t x n 1 Kt 1 a i t n. Let T i be a linear invertible map such that E i a = T i B and let T c be a linear invertible map such that E c a = T c B. Then, as and and E i a = 1 ε n 1 < x, Nx > n+1 κx 1 E c a = 1 + ε n 1 < x, Nx > n+1 κx 1 dett i = 1 ε n 1 < x, Nx > n+1 κx 1, 17 dett c = 1 + ε n 1 < x, Nx > n+1 κx It then follows from property i that there is t such that for all t t = t x T i B T i B t = Ei a E i dett i 1 a i t n a t = 1+ε dett i n p and T c B T c B t = Ec a E c a t Hence 1 a i t n 1 + ε B B t dett c p ε B Bt t t 0 = dett c 1 a c t n and thus for all t t0 = min{t 1, t } n x 1 ε dett c p 1 Kt 1 ε dett c n p. dett i p and 1 ac t n ε dett i p 19 Putting 17 and 18 in 19 we get with new constants c 1 and c for all 1 c 1 ε κ K x p < x, Nx > np 1 16 < x, Nx > n x 1 Kt

17 1 + c ε κ K x p < x, Nx > np 1 ii Now we consider the case that x K is such that κx = 0. Then, by 9, then there is a centered ellipsoid E 0 x that touches K in x and which has at least one axis that is arbitrarily large and there exists a hyperplane H 0 x such that E 0 x H 0 x K H 0 x. Then we continue as in the proof of Lemma 4 and find that for all t small enough n x 1 Kt dett p B Ct dett p 0 < x, Nx > c 1 c where c 1 and c are constants and T is the linear invertible map such that E 0 x = T B. Thus dett = Π n i=1a i, where a i, 1 i n are the lengths of the principal p axes of E 0 x. As dett p = 1, this finishes the proof of the lemma Π n i=1 a i as p > 0 or p < n and as one of the axes is arbitrarily large. 17

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