aa r X i v : . [ m a t h . M G ] F e b New L p Affine Isoperimetric Inequalities ∗ Elisabeth Werner † Deping Ye
Abstract
We prove new L p affine isoperimetric inequalities for all p ∈ [ −∞ , p = − n , a duality formula which shows that L p affine surface areaof a convex body K equals L n p affine surface area of the polar body K ◦ . An affine isoperimetric inequality relates two functionals associated with convex bodies(or more general sets) where the ratio of the functionals is invariant under non-degeneratelinear transformations. These affine isoperimetric inequalities are more powerful thantheir better known Euclidean relatives.This article deals with affine isoperimetric inequalities for the L p affine surface area. L p affine surface area was introduced by Lutwak in the ground breaking paper [26]. Itis now at the core of the rapidly developing L p Brunn Minkowski theory. Contributionshere include new interpretations of L p affine surface areas [32, 37, 38], the discovery ofnew ellipsoids [21, 28], the study of solutions of nontrivial ordinary and, respectively,partial differential equations (see e.g. Chen [9], Chou and Wang [10], Stancu [39, 40]),the study of the L p Christoffel-Minkowski problem by Hu, Ma and Shen [16], a newproof by Fleury, Gu´edon and Paouris [11] of a result by Klartag [18] on concentration ofvolume, and characterization theorems by Ludwig and Reitzner [23].The case p = 1 is the classical affine surface area which goes back to Blaschke [6].Originally a basic affine invariant from the field of affine differential geometry, it hasrecently attracted increased attention too (e.g. [5, 20, 25, 31, 36]). It is fundamental inthe theory of valuations (see e.g. [1, 2, 22, 17]), in approximation of convex bodies bypolytopes [14, 38, 24] and it is the subject of the affine Plateau problem solved in R byTrudinger and Wang [41, 43]. ∗ Keywords: affine surface area, L p Brunn Minkowski theory. 2000 Mathematics Subject Classification:52A20, 53A15 † Partially supported by an NSF grant, a FRG-NSF grant and a BSF grant L p affine isoperimetric inequalities were first established by Lutwak for p > L p affine isoperimetric inequalities for all p ∈ [ −∞ , L p affine surface areas. We establish, for all p = − n , a dualityformula which shows that L p affine surface area of a convex body K equals L n p affinesurface area of the polar body K ◦ . This formula was proved in [15] for p > K in R n isat the origin. We write K ∈ C if K has C boundary with everywhere strictly positiveGaussian curvature. For real p = − n , we define the L p affine surface area as p ( K ) of K as in [26] ( p >
1) and [38] ( p <
1) by as p ( K ) = Z ∂K κ K ( x ) pn + p h x, N K ( x ) i n ( p − n + p dµ K ( x ) (1.1)and as ±∞ ( K ) = Z ∂K κ K ( x ) h x, N K ( x ) i n dµ K ( x ) (1.2)provided the above integrals exist. N K ( x ) is the outer unit normal vector at x to ∂K , theboundary of K . κ K ( x ) is the Gaussian curvature at x ∈ ∂K and µ K denotes the usualsurface area measure on ∂K . h· , ·i is the standard inner product on R n which inducesthe Euclidian norm k · k . In particular, for p = 0 as ( K ) = Z ∂K h x, N K ( x ) i dµ K ( x ) = n | K | , where | K | stands for the n -dimensional volume of K . More generally, for a set M , | M | denotes the Hausdorff content of its appropriate dimension. For p = 1 as ( K ) = Z ∂K κ K ( x ) n +1 dµ K ( x )is the classical affine surface area which is independent of the position of K in space.If the boundary of K is sufficiently smooth then (1.1) and (1.2) can be written asintegrals over the boundary ∂B n = S n − of the Euclidean unit ball B n in R n as p ( K ) = Z S n − f K ( u ) nn + p h K ( u ) n ( p − n + p dσ ( u ) . is the usual surface area measure on S n − . h K ( u ) is the support function of direction u ∈ S n − , and f K ( u ) is the curvature function, i.e. the reciprocal of the Gaussiancurvature κ K ( x ) at this point x ∈ ∂K that has u as outer normal. In particular, for p = ±∞ , as ±∞ ( K ) = Z S n − h K ( u ) n dσ ( u ) = n | K ◦ | (1.3)where K ◦ = { y ∈ R n , h x, y i ≤ , ∀ x ∈ K } is the polar body of K .In Sections 2 and 3 we give new geometric interpretations of the L p affine surfaceareas and obtain as a consequence Corollary 3.1
Let K be a convex body in C and let p = − n be a real number. Then as p ( K ) = as n p ( K ◦ ) . In Section 4 we prove the following new L p affine isoperimetric inequalities. For p ≥ Theorem 4.2
Let K be a convex body with centroid at the origin.(i) If p ≥ , then as p ( K ) as p ( B n ) ≤ (cid:18) | K || B n | (cid:19) n − pn + p , with equality if and only if K is an ellipsoid. For p = 0 , equality holds trivially for all K .(ii) If − n < p < , then as p ( K ) as p ( B n ) ≥ (cid:18) | K || B n | (cid:19) n − pn + p , with equality if and only if K is an ellipsoid.(iii) If K is in addition in C and if p < − n , then c npn + p (cid:18) | K || B n | (cid:19) n − pn + p ≤ as p ( K ) as p ( B n ) . The constant c in (iii) is the constant from the Inverse Santal´o inequality due to Bourgainand Milman [7]. This constant has recently been improved by Kuperberg [19]. We giveexamples that the above isoperimetric inequalities cannot be improved.3n Theorem 4.1 we show a monotonicity behavior of the quotient (cid:16) as r ( K ) n | K | (cid:17) n + rr , namely (cid:18) as r ( K ) n | K | (cid:19) ≤ (cid:18) as t ( K ) n | K | (cid:19) r ( n + t ) t ( n + r ) . and as a consequence obtain Corollary 4.1
Let K be convex body in R n with centroid at the origin.(i) For all p ≥ as p ( K ) as p ( K ◦ ) ≤ n | K | | K ◦ | . (ii) For − n < p < , as p ( K ) as p ( K ◦ ) ≥ n | K | | K ◦ | . If K is in addition in C , inequality (ii) holds for all p < − n . L − nn +2 affine surface area of the polar body It was proved in [32] that for a convex body K ∈ C lim δ → c n | ( K δ ) ◦ | − | K ◦ | δ n +1 = Z S n − dσ ( u ) f K ( u ) n +1 h K ( u ) n +1 = Z ∂K κ K ( x ) n +2 n +1 h x, N K ( x ) i n +1 dµ K ( x )= as − n ( n +2) ( K ) , (2.4)where c n = 2 (cid:16) | B n − | n +1 (cid:17) n +1 and K δ is the convex floating body [36]: The intersection of allhalfspaces H + whose defining hyperplanes H cut off a set of volume δ from K .Assumptions on the boundary of K are needed in order that (2.4) holds.To see that, consider B n ∞ = { x ∈ R n : max ≤ i ≤ n | x i | ≤ } . As κ B n ∞ ( x ) = 0 a.e. on ∂B n ∞ , Z ∂B n ∞ κ B n ∞ ( x ) n +2 n +1 (cid:10) x, N B n ∞ ( x ) (cid:11) n +1 dµ B n ∞ ( x ) = 0 . However lim δ → c n | (( B n ∞ ) δ ) ◦ | − | ( B n ∞ ) ◦ | δ n +1 = ∞ . (2.5)Indeed, writing K for B n ∞ , we will construct a 0-symmetric convex body K such that K δ ⊆ K ⊆ K . Then K ◦ ⊆ K ◦ ⊆ K ◦ δ . Therefore, to show (2.5), it is enough to showthat lim δ → c n | K ◦ | − | K ◦ | δ n +1 = ∞ . R + = { ( x j ) nj =1 : x j ≥ , ≤ j ≤ n } . It is enough to consider K + = R + ∩ K and toconstruct ( K ) + = K ∩ R + .We define ( K ) + to be the intersection of R + with the half-spaces H + i , 1 ≤ i ≤ n +1, where H i = { ( x j ) nj =1 : x i = 1 } , 1 ≤ i ≤ n , and H n +1 = n ( x j ) nj =1 : P nj =1 x j = n − ( n ! δ ) n o , δ > H n +1 (orthogonal to the vector (1 , . . . , δ from K and therefore K δ ⊂ K .Moreover, K ◦ can be written as a convex hull: K ◦ = co (cid:18) {± e i , ≤ i ≤ n } ∪ (cid:26) s ( ε , . . . , ε n ) , ε j = ± , ≤ j ≤ n (cid:27) (cid:19) , where s = n − ( n ! δ ) n . Hence | K ◦ | = 2 n n ! · nn − ( n ! δ ) n and therefore lim δ → | K ◦ | − | K ◦ | δ n +1 = 2 n n ! lim δ → δ − n +1 ( n ! δ ) n ( n − ( n ! δ ) n ) = ∞ . Now we show
Theorem 2.1
Let K be a convex body in C such that ∈ int ( K ) . Then lim δ → c n | ( K δ ) ◦ | − | K ◦ | δ n +1 = as − nn +2 ( K ◦ ) . As a corollary of (2.4) and Theorem 2.1 we get that for a convex body K ∈ C as − n ( n +2) ( K ) = as − nn +2 ( K ◦ ) . (2.6)This is a special case for p = − n ( n + 2) of the formula as p ( K ) = as n p ( K ◦ ) proved in [15]for p >
0. We will show in the next section that this formula holds for all p < , p = − n for convex bodies with sufficiently smooth boundary. For p = 0 (and K ∈ C ) theformula holds trivially as as ( K ) = n | K | and as ∞ ( K ◦ ) = n | K | (see [38]).For the proof of Theorem 2.1 we need the following lemmas. Lemma 2.1
Let K ∈ C . Then for any x ∈ ∂K ◦ , we have lim δ → h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) = h x, N K ◦ ( x ) i c n ( κ K ◦ ( x )) n +1 where x δ ∈ ∂ ( K δ ) ◦ is in the ray passing through and x . roof Since K , and hence also K δ , are in C one has that K ◦ and ( K δ ) ◦ are in C . Therefore,for x ∈ ∂K ◦ there exists a unique y ∈ ∂K , such that, h x, y i = 1, namely y = N K o ( x ) h N K o ( x ) , x i . y has outer normal vector N K ( y ) = x k x k and k x k = h y, N K ( y ) i .Similarly, for x δ ∈ ∂ ( K δ ) ◦ there exists a unique y δ in ∂K δ such that h x δ , y δ i = 1,namely y δ = N ( K δ ) o ( x δ ) h N ( K δ ) o ( x δ ) , x δ i , y δ has outer normal vector N K δ ( y δ ) = x δ k x δ k = x k x k and k x δ k = h y δ , N K δ ( y δ ) i .Let y ′ = [0 , y ] ∩ ∂K δ ([ z , z ] denotes the line segment from z to z ) and let y ′ δ ∈ ∂K be such that y δ = [0 , y ′ δ ] ∩ K δ .We have 1 k x k = h y, N K ( y ) i ≥ h y ′ δ , N K ( y ) i = h y ′ δ , x k x k i , k x δ k = h y δ , N K δ ( y δ ) i ≥ h y ′ , N K δ ( y δ ) i = h y ′ , x k x k i . Hence (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) = (cid:20)(cid:18) h y, N K ( y ) ih y δ , N K δ ( y δ ) i (cid:19) n − (cid:21) ≥ " h y ′ δ , x k x k ih y δ , x k x k i ! n − = (cid:20)(cid:18) k y ′ δ kk y δ k (cid:19) n − (cid:21) , (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) = (cid:20)(cid:18) h y, N K ( y ) ih y δ , N K δ ( y δ ) i (cid:19) n − (cid:21) ≤ " h y, x k x k ih y ′ , x k x k i ! n − = (cid:20)(cid:18) k y kk y ′ k (cid:19) n − (cid:21) (2.7)and therefore h x, N K ◦ ( x ) i n (cid:20)(cid:18) k y ′ δ kk y δ k (cid:19) n − (cid:21) ≤ h x, N K ◦ ( x ) i n (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) ≤ h x, N K ◦ ( x ) i n (cid:20)(cid:18) k y kk y ′ k (cid:19) n − (cid:21) . We first consider the lower bound.lim δ → h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) ≥ lim δ → h x, N K ◦ ( x ) ih y ′ δ , N K δ ( y ′ δ ) i h y ′ δ , N K δ ( y ′ δ ) i n δ n +1 (cid:20)(cid:18) k y ′ δ kk y δ k (cid:19) n − (cid:21) . As δ → y ′ δ → y . As K is in C , N K δ ( y ′ δ ) → N K ( y ) as δ → δ → h y ′ δ , N K δ ( y ′ δ ) i = h y, N K ( y ) i . By Lemma 7 and Lemma 10 of [36],lim δ → h y ′ δ , N K δ ( y ′ δ ) i n δ n +1 (cid:20)(cid:18) k y ′ δ kk y δ k (cid:19) n − (cid:21) = ( κ K ( y )) n +1 c n . Hence lim δ → h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) ≥ h x, N K ◦ ( x ) ih y, N K ( y ) i ( κ K ( y )) n +1 c n = h x, N K ◦ ( x ) i c n ( κ K ◦ ( x )) n +1 . The last equation follows from the fact that if K ∈ C , then, for any y ∈ ∂K , there is aunique point x ∈ ∂K ◦ such that h x, y i = 1 and [15] h y, N K ( y ) ih x, N K ◦ ( x ) i = ( κ K ( y ) κ K ◦ ( x )) n +1 . (2.8)Similarly, one gets for the upper boundlim δ → h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) ≤ h x, N K ◦ ( x ) i c n ( κ K ◦ ( x )) n +1 , hence altogether lim δ → h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) = h x, N K ◦ ( x ) i c n ( κ K ◦ ( x )) n +1 . Lemma 2.2
Let K ∈ C . Then we have h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) ≤ c ( K, n ) , where c ( K, n ) is a constant (depending on K and n only) and x and x δ are as in Lemma2.1. Proof
By (2.7) h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) ≤ h x, N K ◦ ( x ) ih y, N K ( y ) i h y, N K ( y ) i n δ n +1 (cid:20)(cid:18) k y kk y ′ k (cid:19) n − (cid:21) ≤ h x, N K ◦ ( x ) ih y, N K ( y ) i (cid:18) k y kk y ′ k (cid:19) n h y, N K ( y ) i n δ n +1 (cid:20) − (cid:18) k y ′ kk y k (cid:19) n (cid:21) . K δ is increasing to K as δ →
0, there exists δ > δ < δ ,0 ∈ int( K δ ). Therefore there exits α > B n (0 , α ) ⊂ K δ ⊂ K ⊂ B n (0 , α ) forall δ < δ . B n (0 , r ) is the n -dimensional Euclidean ball centered at 0 with radius r .Hence for δ < δ h x, N K ◦ ( x ) i n δ n +1 (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) ≤ α − n +1) h y, N K ( y ) i n δ n +1 (cid:20) − (cid:18) k y ′ kk y k (cid:19) n (cid:21) ≤ C ′ r ( y ) − n − n +1 due to Lemma 6 in [36]. Here r ( y ) is the radius of the biggest Euclidean ball containedin K and touching ∂K at y .Since K is C , by the Blaschke rolling theorem (see [34]) there is r > r ≤ min y ∈ ∂K r ( y ). We put c ( K, n ) = C ′ r − n − n +1 . Proof of Theorem 2.1. | ( K δ ) ◦ | − | K ◦ | δ n +1 = 1 n δ n +1 Z ∂K o h x, N K ◦ ( x ) i (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) dµ K ◦ ( x ) . Combining Lemma 2.1, Lemma 2.2 and Lebesgue’s convergence theorem, gives Theorem2.1: lim δ → | ( K δ ) ◦ | − | K ◦ | δ n +1 = lim δ → n δ n +1 Z ∂K o h x, N K ◦ ( x ) i (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) dµ K ◦ ( x )= Z ∂K o lim δ → n δ n +1 h x, N K ◦ ( x ) i (cid:20)(cid:18) k x δ kk x k (cid:19) n − (cid:21) dµ K ◦ ( x )= Z ∂K o h x, N K ◦ ( x ) i c n ( κ K ◦ ( x )) n +1 dµ K ◦ ( x )= 1 c n as − nn +2 ( K ◦ ) . Remark
The proof of Theorem 2.1 provides a uniform method to evaluatelim t → | ( K t ) ◦ | − | K ◦ | t n +1 where K t is a family convex bodies constructed from the convex body K such that K t ⊂ K or- similarly- such that K ⊂ K t . In particular, we can apply this method to prove theanalog statements as in (2.4) and Theorem 2.1 if we take as K t the illumination body of K [42], or the Santal´o body of K [31], or the convolution body of K [33] - and there aremany more. 8 L p affine surface areas We now prove that for all p = − n and all K ∈ C , as p ( K ) = as n p ( K ◦ ). To do so, weuse the surface body of a convex body which was introduced in [37, 38]. We also give anew geometric interpretation of L p affine surface area for all p = − n . Definition 3.1
Let s ≥ and f : ∂K → R be a nonnegative, integrable function.The surface body K f,s is the intersection of all the closed half-spaces H + whose defininghyperplanes H cut off a set of f µ K -measure less than or equal to s from ∂K . Moreprecisely, K f,s = \ R ∂K ∩ H − fdµ K ≤ s H + . Theorem 3.1
Let K be a convex body in C and such that is the center of gravity of K . Let f : ∂K → R be an integrable function such that f ( x ) > c for all x ∈ ∂K andsome constant c > . Let β n = 2 (cid:0) | B n − | (cid:1) n − . Then lim s → β n | ( K f,s ) ◦ | − | K ◦ | s n − = Z S n − dσ ( u ) h K ( u ) n +1 f K ( u ) n − (cid:0) f ( N − K ( u )) (cid:1) n − where N K : ∂K → S n − , x → N K ( x ) = u is the Gauss map. Proof
Let u ∈ S n − . Let x ∈ ∂K be such that N K ( x ) = u and let x s ∈ ∂K f,s be such that N K f,s ( x s ) = u . Let H ∆ = H ( x − ∆ u, u ) be the hyperplane through x − ∆ u with outernormal vector u . Since K has everywhere strictly positive Gaussian curvature, by Lemma21 in [38] almost everywhere on ∂K ,lim ∆ → | ∂K ∩ H − ∆ | Z ∂K ∩ H − ∆ | f ( x ) − f ( y ) | dµ K ( y ) = 0 . This implies that lim ∆ → | ∂K ∩ H − ∆ | Z ∂K ∩ H − ∆ f ( y ) dµ ∂K ( y ) = f ( x ) . (3.9)Let b s = h K ( u ) − h K f,s ( u ). As H ( x − b s u, u ) = H ( x s , u ) (the hyperplane through x s withouter normal u ) and as b s → s →
0, (3.9) implieslim s → | ∂K ∩ H − ( x s , u ) | Z ∂K ∩ H − ( x s ,u ) f ( y ) dµ K ( y ) = f ( x ) . (3.10)9ence there exists s small enough, such that for all s < s , s ≤ Z ∂K ∩ H − ( x s ,u ) f ( y ) dµ K ( y ) ≤ (1 + ε ) f ( x ) | ∂K ∩ H − ( x s , u ) | . (3.11)As ∂K has everywhere strictly positive Gaussian curvature, the indicatrix of Dupinexists everywhere on ∂K and is an ellipsoid. It then follows from (3.11) with Lemmas1.2, 1.3 and 1.4 in [37] that there exists 0 < s < s such that for all 0 < s < s s ≤ (1 + ε ) f ( x ) | B n − | p f K ( u ) (2 b s ) n − , or, equivalently b s s n − ≥ − c εβ n f ( N − K ( u )) n − f K ( u ) n − , (3.12)where c is an absolute constant.Let now x ′ s ∈ [0 , x ] ∩ ∂K f,s . Then h x ′ s , u i ≤ h K f,s ( u ). Therefore b s = h K ( u ) − h K f,s ( u ) ≤h x − x ′ s , u i .Hence for s sufficiently small b s s n − ≤ h x − x ′ s , u i s n − ≤ h x, u i s n − (cid:18) − k x ′ s kk x k (cid:19) ≤ h x, u i s n − k x ′ s − x kk x k≤ (1 + ε ) h x, N K ( x ) i n s n − (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) . (3.13)The last inequality follows as 1 − (cid:16) k x ′ s kk x k (cid:17) n ≥ (1 − ε ) n k x ′ s − x kk x k for sufficiently small s . ByLemma 23 in [38]lim s → ns n − h x, N K ( x ) i (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) = 1 β n f ( N − K ( u )) n − f K ( u ) n − . (3.14)Thus we get from (3.12), (3.13) and (3.14) thatlim s → b s s n − = 1 β n f ( N − K ( u )) n − f K ( u ) n − . (3.15)As (1 − t ) − n ≥ nt for 0 ≤ t < s → β n ns n − (cid:0) [ h K f,s ( u )] − n − [ h K ( u )] − n (cid:1) = lim s → β n ns n − [ h K ( u )] − n "(cid:18) b s h K ( u ) (cid:19) − n − ≥ lim s → β n [ h K ( u )] n +1 b s s n − = 1[ h K ( u )] n +1 f ( N − K ( u )) n − f K ( u ) n − . (3.16)10s h K f,s ( u ) ≥ h x ′ s , u i , h K f,s ( u ) h K ( u ) ≥ h x ′ s , u ih x, u i = k x ′ s kk x k . (3.17)Since K ∈ C , h K f,s ( u ) → h K ( u ) as s →
0. Therefore,lim s → β n ns n − (cid:0) [ h K f,s ( u )] − n − [ h K ( u )] − n (cid:1) = lim s → β n ns n − [ h K f,s ( u )] − n (cid:20) − (cid:18) h K f,s ( u ) h K ( u ) (cid:19) n (cid:21) ≤ lim s → β n ns n − [ h K f,s ( u )] − n (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) = lim s → β n ns n − [ h K f,s ( u )] − n h x, u ih x, u i (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) = lim s → h K ( u ) [ h K f,s ( u )] n lim s → β n ns n − h x, N K ( x ) i (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) = 1[ h K ( u )] n +1 f ( N − K ( u )) n − f K ( u ) n − (3.18)where the last equality follows from (3.14).Altogether, (3.16) and (3.18) givelim s → β n ns n − (cid:0) [ h K f,s ( u )] − n − [ h K ( u )] − n (cid:1) = 1[ h K ( u )] n +1 f ( N − K ( u )) n − f K ( u ) n − . Thereforelim s → β n | ( K f,s ) ◦ | − | K ◦ | s n − = lim s → β n n s n − Z S n − (cid:20)(cid:18) h K f,s ( u ) (cid:19) n − (cid:18) h K ( u ) (cid:19) n (cid:21) dσ ( u )= Z S n − lim s → β n n s n − (cid:20)(cid:18) h K f,s ( u ) (cid:19) n − (cid:18) h K ( u ) (cid:19) n (cid:21) dσ ( u )= Z S n − dσ ( u ) h K ( u ) n +1 f K ( u ) n − (cid:0) f ( N − K ( u )) (cid:1) n − , provided we can interchange integration and limit.We show this next. To do so, we show that for all u ∈ S n − and all sufficiently small s >
0, 1 n s n − (cid:20)(cid:18) h K f,s ( u ) (cid:19) n − (cid:18) h K ( u ) (cid:19) n (cid:21) ≤ g ( u )11ith R S n − g ( u ) dσ ( u ) < ∞ . As 0 ∈ int ( K ), the interior of K , there exists α > s sufficiently small B n (0 , α ) ⊂ K f,s ⊂ K ⊂ B n (0 , α ). Therefore, α ≤ h K f,s ( u ) ≤ h K ( u ) ≤ α and α ≤ h K ( u ) ≤ h Kf,s ( u ) ≤ α .With (3.17), we thus get for all s > n s n − (cid:18)(cid:0) h K f,s ( u ) (cid:1) − n − (cid:0) h K ( u ) (cid:1) − n (cid:19) = 1 n s n − (cid:0) h K f,s ( u ) (cid:1) − n (cid:18) − (cid:0) h K f,s ( u ) (cid:1) n (cid:0) h K ( u ) (cid:1) n (cid:19) ≤ α − n n s n − (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) ≤ α − ( n +1) h x, u i n s n − (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) . By Lemma 17 in [38] there exists s such that for all s ≤ s h x, u i s n − (cid:20) − (cid:18) k x ′ s kk x k (cid:19) n (cid:21) ≤ C ( M f ( x )) n − r ( x ) , where C is an absolute constant and, as in the proof of Lemma 2.2, r ( x ) is the biggestEuclidean ball contained in K that touches ∂K at x . Thus, as ∂K is C , by Blaschke’srolling theorem (see [34]) there is r such that r ( x ) ≥ r . M f ( x ) = inf
Let K be a convex body in C and such that is the center of gravity of K . Let f : ∂K → R be an integrable function such that f ( y ) > c for all y ∈ ∂K andsome constant c > . Let β n = 2 (cid:0) | B n − | (cid:1) n − . Then lim s → β n | ( K f,s ) ◦ | − | K ◦ | s n − = Z ∂K ◦ (cid:18) h x, N K ◦ ( x ) ih y ( x ) , N K ( y ( x )) i (cid:19) κ K ( y ( x )) n − f ( y ( x )) n − ! dµ K ◦ ( x ) Here y ( x ) ∈ ∂K is such that h y ( x ) , x i = 1 . roof We follow the pattern of the proof of Theorem 3.1 integrating now over ∂K ◦ insteadof S n − .As a corollary we get the following geometric interpretation of L p affine surface area. Corollary 3.1
Let K ∈ C be a convex body. For p ∈ R , p = − n , let f p : ∂K → R bedefined by f p ( y ) = κ K ( y ) n p n + p ) h y, N K ( y ) i − ( n − n n + p )2( n + p ) . Then(i) lim s → β n | ( K f p ,s ) ◦ | − | K ◦ | s n − = as n p ( K ◦ ) . (ii) lim s → β n | ( K f p ,s ) ◦ | − | K ◦ | s n − = as p ( K ) . (iii) as p ( K ) = as n p ( K ◦ ) . Proof
Notice first that f p ( y ) verifies the conditions of Theorems 3.1 and 3.2.(i) For x ∈ ∂K ◦ , let now y ( x ) be the (unique) element in ∂K such that h x, y ( x ) i = 1.Then, by Theorem 3.2, with f ( y ( x )) = f p ( y ( x )), and with (2.8)lim s → β n | (cid:0) K f p ,s (cid:1) ◦ | − | K ◦ | s n − = Z ∂K ◦ h x, N K ◦ ( x ) i h y ( x ) , N K ( y ( x )) i n ( n +1) n + p κ K ( y ( x )) nn + p dµ K ◦ ( x )= Z ∂K ◦ κ K ◦ ( x ) nn + p h x, N K ◦ ( x ) i n − pn + p dµ K ◦ ( x ) = as n p ( K ◦ ) . (ii) For u ∈ S n − , let now y ∈ ∂K be such that N K ( y ) = u . Then f p ( N − K ( u )) = f K ( u ) − n p n + p ) h K ( u ) − ( n − n n + p )2( n + p ) . By Theorem 3.1 with f ( N − K ( u )) = f p ( N − K ( u ))lim s → β n | (cid:0) K f p ,s (cid:1) ◦ | − | K ◦ | s n − = Z S n − f K ( u ) nn + p h K ( u ) n ( p − n + p dσ ( u ) = as p ( K ) . (iii) follows from (i) and (ii). 13 Inequalities
Theorem 4.1
Let s = − n, r = − n, t = − n be real numbers. Let K be a convex body in R n with centroid at the origin and such that µ K { x ∈ ∂K : κ K ( x ) = 0 } = 0 .(i) If ( n + r )( t − s )( n + t )( r − s ) > , then as r ( K ) ≤ (cid:0) as t ( K ) (cid:1) ( r − s )( n + t )( t − s )( n + r ) (cid:0) as s ( K ) (cid:1) ( t − r )( n + s )( t − s )( n + r ) . (ii) If ( n + r ) t ( n + t ) r > , then (cid:18) as r ( K ) n | K | (cid:19) ≤ (cid:18) as t ( K ) n | K | (cid:19) r ( n + t ) t ( n + r ) . Proof (i) By H¨older’s inequality -which enforces the condition ( n + r )( s − t )( n + t )( s − r ) > as r ( K ) = Z ∂K κ K ( x ) rn + r h x, N K ( x ) i n ( r − n + r dµ K ( x )= Z ∂K κ K ( x ) tn + t h x, N K ( x ) i n ( t − n + t ! ( r − s )( n + t )( t − s )( n + r ) κ K ( x ) sn + s h x, N K ( x ) i n ( s − n + s ! ( t − r )( n + s )( t − s )( n + r ) dµ K ( x ) ≤ (cid:0) as t ( K ) (cid:1) ( r − s )( n + t )( t − s )( n + r ) (cid:0) as s ( K ) (cid:1) ( t − r )( n + s )( t − s )( n + r ) . (ii) Similarly, again using H¨older’s inequality -which now enforces the condition ( n + r ) t ( n + t ) r > ,as r ( K ) = Z ∂K κ K ( x ) rn + r h x, N K ( x ) i n ( r − n + r dµ K ( x ) = Z ∂K κ K ( x ) tn + t h x, N K ( x ) i n ( t − n + t ! r ( n + t ) t ( n + r ) dµ K ( x ) h x, N K ( x ) i ( r − t ) n ( n + r ) t ≤ (cid:0) as t ( K ) (cid:1) r ( n + t ) t ( n + r ) (cid:0) n | K | (cid:1) ( t − r ) n ( n + r ) t . Condition ( n + r )( t − s )( n + t )( r − s ) > − n < s < r < t , s < − n < t < r , r < t < − n < s , t < r < s < − n , s < r < t < − n , r < s < − n < t , t < − n < s < r and − n < t < r < s . 14ote also that (ii) describes a monotonicity condition for (cid:16) as r ( K ) n | K | (cid:17) n + rr : if 0 < r < t ,or r < t < − n , or − n < r < t < (cid:18) as r ( K ) n | K | (cid:19) n + rr ≤ (cid:18) as t ( K ) n | K | (cid:19) n + tt . We now analyze various subcases of Theorem 4.1 (i) and (ii). For r = 0, if n ( s − t ) s ( n + t ) > n | K | ≤ (cid:0) as t ( K ) (cid:1) s ( n + t ) n ( s − t ) (cid:0) as s ( K ) (cid:1) t ( n + s ) n ( t − s ) . For s = 0, if t ( n + r ) r ( n + t ) > as r ( K ) ≤ ( n | K | ) n ( t − r ) t ( n + r ) (cid:0) as t ( K ) (cid:1) r ( n + t ) t ( n + r ) . (4.19)For s → ∞ , if n + rn + t > as r ( K ) ≤ (cid:0) as ∞ ( K ) (cid:1) r − tn + r (cid:0) as t ( K ) (cid:1) n + tn + r . (4.20)For r → ∞ , if t − sn + t > K is in C , as ∞ ( K ) = n | K ◦ | ≤ (cid:0) as t ( K ) (cid:1) n + tt − s (cid:0) as s ( K ) (cid:1) n + ss − t . (4.21)As for all convex bodies K , as ∞ ( K ) ≤ n | K ◦ | (see [38]), it follows from (4.19) that,for all convex body K with centroid at origin, as r ( K ) ≤ ( n | K | ) nn + r ( n | K ◦ | ) rn + r , r > n | K | (cid:0) n | K ◦ | (cid:1) tn ≤ (cid:0) as t ( K ) (cid:1) n + tn , − n < t < . (4.23)Similarly, (4.21) implies that, if in addition K is in C , n | K ◦ | (cid:0) n | K | (cid:1) nt ≤ (cid:0) as t ( K ) (cid:1) n + tt , t < − n (4.24)(4.22) can also be obtained from Proposition 4.6 of [26] and Theorem 3.2 of [15].Inequalities (4.22), (4.23) and (4.24) yield the following Corollary which was provedby Lutwak [26] in the case p ≥
1. 15 orollary 4.1
Let K be convex body in R n with centroid at the origin.(i) For all p ≥ as p ( K ) as p ( K ◦ ) ≤ n | K | | K ◦ | . (ii) For − n < p < , as p ( K ) as p ( K ◦ ) ≥ n | K | | K ◦ | . If K is in addition in C , inequality (ii) holds for all p < − n . Thus, using Santal´o inequality in (i), for p ≥ as p ( K ) as p ( K ◦ ) ≤ as p ( B n ) , and inverseSantal´o inequality in (ii), for − n < p < as p ( K ) as p ( K ◦ ) ≥ c n as p ( B n ) . c is the constantin the inverse Santal´o inequality [7, 19]. Proof (i) follows immediately form (4.22). (ii) follows from (4.23) if − n < p < p < − n .Lutwak [26] proved for p ≥ as p ( K ) as p ( B n ) ≤ (cid:18) | K || B n | (cid:19) n − pn + p with equality if and only if K is an ellipsoid. We now generalize these L p -affine isoperi-metric inequalities to p < Theorem 4.2
Let K be a convex body with centroid at the origin.(i) If p ≥ , then as p ( K ) as p ( B n ) ≤ (cid:18) | K || B n | (cid:19) n − pn + p , with equality if and only if K is an ellipsoid. For p = 0 , equality holds trivially for all K .(ii) If − n < p < , then as p ( K ) as p ( B n ) ≥ (cid:18) | K || B n | (cid:19) n − pn + p , with equality if and only if K is an ellipsoid.(iii) If K is in addition in C and if p < − n , then as p ( K ) as p ( B n ) ≥ c npn + p (cid:18) | K || B n | (cid:19) n − pn + p . where c is the constant in the inverse Santal´o inequality [7, 19].
16e cannot expect to get a strictly positive lower bound in Theorem 4.2 (i), even if K is in C : Consider, in R , the convex body K ( R, ε ) obtained as the intersection of fourEuclidean balls with radius R centered at ( ± ( R − , , ± ( R − R arbitrarily large.To obtain a body in C , we “round” the corners by putting there Euclidean balls withradius ε , ε arbitrarily small. Then as p ( K ( R, ε )) ≤ R p p + 4 π ε p . A similar constructioncan be done in higher dimensions.This example also shows that, likewise, we cannot expect finite upper bounds inTheorem 4.2 (ii) and (iii). If − < p <
0, then as p ( K ( R, ε )) ≥ p +1)2+ p R − p p . If p < − − < p < as p ( K ( R, ε ) ◦ ) = as p ( K ( R, ε )) ≥ R − p +2 p p . Note also that in part (iii) we cannot remove the constant c npn + p . Indeed, if p → −∞ ,the inequality becomes c n | B n | ≤ | K || K ◦ | . Proof of Theorem 4.2 (i) The case p = 0 is trivial. We prove the case p >
0. Combining inequality (4.22),the Blaschke Santal´o inequality, and as q ( B n ) = n | B n | nn + q | B n | qn + q , one obtains as p ( K ) as p ( B n ) ≤ (cid:18) | K ◦ || B n | (cid:19) pn + p (cid:18) | K || B n | (cid:19) nn + p ≤ (cid:18) | K || B n | (cid:19) n − pn + p . This proves the inequality. The equality case follows from the equality case for theBlaschke Santal´o inequality.(ii) Combining inequality (4.23) and (cid:0) as p ( B n ) (cid:1) n + pn = n | B n | (cid:0) n | B n | (cid:1) pn , one gets, for − n < p < (cid:18) as p ( K ) as p ( B n ) (cid:19) n + pn ≥ (cid:18) | K || B n | (cid:19) (cid:18) | K ◦ || B n | (cid:19) pn ≥ (cid:18) | K || B n | (cid:19) n − pn where the last inequality follows from the Blaschke Santal´o inequality. As pn < | K | | K ◦ | ) pn ≥ ( | B n | | B n | ) pn . As n + p > as p ( K ) as p ( B n ) ≥ (cid:18) | K || B n | (cid:19) n − pn + p . The equality case follows from the equality case for the Blaschke Santal´o inequality.17iii) Similarly, combining (4.24), n | B n | = ( as p ( B n )) n + pp − ( as ( B n )) n +11 − p , and the InverseSantal´o inequality, we get, for p < − n , (cid:18) as p ( K ) as p ( B n ) (cid:19) n + pp ≥ (cid:18) | K ◦ || B n | (cid:19) (cid:18) | K || B n | (cid:19) np ≥ c n (cid:18) | K || B n | (cid:19) n − pp . As n + pp > as p ( K ) as p ( B n ) ≥ c npn + p (cid:18) | K || B n | (cid:19) n − pn + p . The L − n affine surface area was defined in [32] for convex bodies K in C and withcentroid at the origin by as − n ( K ) = max u ∈ S n − f K ( u ) h K ( u ) n +12 . More generally, one could define the L − n affine surface area for any convex body K with centroid at the origin by as − n ( K ) = sup x ∈ ∂K h x,N K ( x ) i n +12 κ K ( x ) . But as in most cases then as − n ( K ) = ∞ , it suffices to consider K in C .A statement similar to Theorem 4.1 holds. Proposition 4.1
Let K be a convex body in C with centroid at the origin. Let p = − n and s = − n be real numbers.(i) If n ( s − p )( n + p )( n + s ) ≥ , then as p ( K ) ≤ (cid:0) as − n ( K ) (cid:1) n ( s − p )( n + p )( n + s ) as s ( K ) . (ii) If n ( s − p )( n + p )( n + s ) ≤ , then as p ( K ) ≥ (cid:0) as − n ( K ) (cid:1) n ( s − p )( n + p )( n + s ) as s ( K ) . (iii) The L − n affine isoperimetric inequality holds as − n ( K ) as − n ( B n ) ≥ | K || B n | . Proof (i) and (ii) as p ( K ) = Z ∂K κ K ( x ) pn + p h x, N K ( x ) i n ( p − n + p dµ K ( x )= Z ∂K κ K ( x ) sn + s h x, N K ( x ) i n ( s − n + s ! h x, N K ( x ) i n +12 κ K ( x ) ! n ( s − p )( n + p )( n + s ) dµ K ( x )18hich is ≤ (cid:0) as − n ( K ) (cid:1) n ( s − p )( n + p )( n + s ) as s ( K ) , if n ( s − p )( n + p )( n + s ) ≥ ≥ (cid:0) as − n ( K ) (cid:1) n ( s − p )( n + p )( n + s ) as s ( K ) , if n ( s − p )( n + p )( n + s ) ≤ n ( s − p )( n + p )( n + s ) > s > p > − n or p < s < − n or s < − n < p .If p = 0 and s → ∞ , then as − n ( K ) ≥ s | K || K ◦ | . (4.25)This gives the L − n affine isoperimetric inequality as − n ( K ) as − n ( B n ) = as − n ( K ) ≥ s | K || K ◦ | ≥ s | K | | K | | K ◦ | ≥ s | K | | B n | = | K || B n | . Analogous to corollary 4.1, an immediate consequence of (4.25) is the following corol-lary. It can also be proved directly using (2.8).
Corollary 4.2
Let K be a convex body in C with centroid at the origin. Then as − n ( K ◦ ) as − n ( K ) ≥ as − n ( B n ) . Acknowledgment
The authors would like to thank the referee for the many helpfulsuggestions.
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Proceedings of the International Congressof Mathematicians, vol.III, 221-231, Beijing, (2002).Elisabeth WernerDepartment of Mathematics Universit´e de Lille 1Case Western Reserve University UFR de Math´ematiqueCleveland, Ohio 44106, U. S. A. 59655 Villeneuve d’Ascq, France [email protected]
Deping YeDepartment of MathematicsCase Western Reserve UniversityCleveland, Ohio 44106, U. S. A. [email protected]@case.edu