Estimates for moments of general measures on convex bodies
aa r X i v : . [ m a t h . M G ] D ec ESTIMATES FOR MOMENTS OF GENERAL MEASURES ONCONVEX BODIES
SERGEY BOBKOV, BO’AZ KLARTAG, AND ALEXANDER KOLDOBSKY
Abstract.
For p ≥ n ∈ N , and an origin-symmetric convex body K in R n , let d ovr ( K, L np ) = inf (cid:26)(cid:16) | D || K | (cid:17) /n : K ⊆ D, D ∈ L np (cid:27) be the outer volume ratio distance from K to the class L np of the unit balls of n -dimensional subspaces of L p . We prove that there exists an absolute constant c > c √ n √ p log log n ≤ sup K d ovr ( K, L np ) ≤ √ n. This result follows from a new slicing inequality for arbitrary measures, in thespirit of the slicing problem of Bourgain. Namely, there exists an absoluteconstant
C > p ≥ , any n ∈ N , any compact set K ⊆ R n ofpositive volume, and any Borel measurable function f ≥ K ,(0.2) Z K f ( x ) dx ≤ C √ p d ovr ( K, L np ) | K | /n sup H Z K ∩ H f ( x ) dx, where the supremum is taken over all affine hyperplanes H in R n . Combin-ing (0.2) with a recent counterexample for the slicing problem with arbitrarymeasures from [9], we get the lower estimate from (0.1).In turn, inequality (0.2) follows from an estimate for the p -th absolutemoments of the function f min ξ ∈ S n − Z K | ( x, ξ ) | p f ( x ) dx ≤ ( Cp ) p/ d p ovr ( K, L np ) | K | p/n Z K f ( x ) dx. Finally, we prove a result of the Busemann-Petty type for these moments. Introduction
Suppose that K ⊆ R n ( n ≥
1) is a centrally-symmetric convex set of volume one(i.e., K = − K ). Given an even continuous probability density f : K → [0 , ∞ ), and p ≥
1, can we find a direction ξ such that the p -th absolute moment(1.1) M K,f,p ( ξ ) = Z K | ( x, ξ ) | p f ( x ) dx is smaller than a constant which does not depend on K and f ? More precisely andin a more relaxed form, let γ ( p, n ) be the smallest number γ > ξ ∈ S n − M K,f,p ( ξ ) ≤ γ p | K | p/n Z K f ( x ) dx This material is based upon work supported by the US National Science Foundation underGrant DMS-1440140 while the authors were in residence at the Mathematical Sciences ResearchInstitute in Berkeley, California, during the Fall 2017 semester. The first- and third-named authorswere supported in part by the NSF Grants DMS-1612961 and DMS-1700036. The second-namedauthor was supported in part by a European Research Council (ERC) grant. for all centrally-symmetric convex bodies K ⊆ R n and all even continuous func-tions f ≥ K . Here and below, we denote by S n − = { ξ ∈ R n : | ξ | = 1 } theEuclidean unit sphere centered at the origin, and | K | stands for volume of appro-priate dimension. (Note that the continuity property of f in the definition 1.1 isirrelevant and may easily be replaced by measurability.) As we will see, there is atwo-sided bound on γ ( p, n ). Theorem 1.1.
With some positive absolute constants c and C , for any p ≥ , c √ n √ log log n ≤ γ ( p, n ) ≤ C √ pn. To describe the way the upper bound is obtained, denote by L np the class of theunit balls of n -dimensional subspaces of L p . Equivalently (see [11, p. 117]), L np isthe class of all centrally-symmetric convex bodies D in R n such that there exists afinite Borel measure ν D on S n − satisfying(1.3) k x k pD = Z S n − | ( x, θ ) | p dν D ( θ ) , ∀ x ∈ R n . Here k x k D = inf { a ≥ x ∈ aD } is the norm generated by D . Note that L n = Π ∗ n is the class of polar projection bodies which, in particular, contains thecross-polytopes; see [11, Ch.8] for details.For a (bounded) set K in R n , define the quantity V ( K, L np ) = inf (cid:8) | D | /n : K ⊆ D, D ∈ L np (cid:9) . If K is measurable and has positive volume, we have the relation V ( K, L np ) = d ovr ( K, L np ) | K | /n , with(1.4) d ovr ( K, L np ) = inf (cid:26)(cid:16) | D || K | (cid:17) /n : K ⊆ D, D ∈ L np (cid:27) . For convex K , the latter may be interpreted as the outer volume ratio distance from K to the class of unit balls of n -dimensional subspaces of L p . The next body-wiseestimates refine the upper bound in Theorem 1.1 in terms of the d ovr -distance. Theorem 1.2.
Given a probability measure µ on R n with a compact support K ,for every p ≥ , min ξ ∈ S n − (cid:16) Z | ( x, ξ ) | p dµ ( x ) (cid:17) /p ≤ C √ p V ( K, L np ) , where C is an absolute constant. In particular, if f is a non-negative continuousfunction on a compact set K ⊆ R n of positive volume, then min ξ ∈ S n − M K,f,p ( ξ ) ≤ ( Cp ) p/ d p ovr ( K, L np ) | K | p/n Z K f ( x ) dx. In the class of centrally-symmetric convex bodies K in R n , there is a dimen-sional bound d ovr ( K, L np ) ≤ √ n , which follows from John’s theorem and the factthat ellipsoids belong to L np for all p ≥ d ovr ( K, L np ) is bounded by absolute constants. These classes STIMATES FOR MOMENTS OF GENERAL MEASURES ON CONVEX BODIES 3 include duals of bodies with bounded volume ratio (see [14]) and the unit balls ofnormed spaces that embed in L q , 1 ≤ q < ∞ (see [18, 15]). In the case p = 1,they also include all unconditional convex bodies [14]. The proofs in these papersestimate the distance from the class of intersection bodies, but the actual bodiesused there (the Euclidean ball for p > p = 1) alsobelong to the classes L np , so the same arguments work for L np .In order to prove the lower estimate of Theorem 1.1, we first establish the con-nection between question (1.1) and the slicing problem for arbitrary measures. Theslicing problem of Bourgain [2, 3] asks whether sup n L n < ∞ , where L n is theminimal positive number L such that, for any centrally-symmetric convex body K ⊆ R n , | K | ≤ L max ξ ∈ S n − | K ∩ ξ ⊥ | | K | /n . Here, ξ ⊥ is the hyperplane in R n passing through the origin and perpendicular to thevector ξ , and we write | K ∩ ξ ⊥ | for the ( n − L n ≤ Cn / was established bythe second-named author [8], removing a logarithmic term from an earlier estimateby Bourgain [4].The slicing problem for arbitrary measures was introduced in [12] and consideredin [13, 14, 15, 5, 9]. In analogy with the original problem, for a centrally-symmetricconvex body K ⊆ R n , let S n,K be the smallest positive number S satisfying(1.5) Z K f ( x ) dx ≤ S max ξ ∈ S n − Z K ∩ ξ ⊥ f ( x ) dx | K | n for all even continuous functions f ≥ R n (where dx on the right-hand siderefers to the Lebesgue measure on the corresponding affine subspace of R n ). It wasproved in [13] that S n = sup K ⊆ R n S n,K ≤ √ n. However, for many classes of bodies, including intersection bodies [12] and uncondi-tional convex bodies [14], the quantity S n,K turns out to be bounded by an absoluteconstant. In particular, if K is the unit ball of an n -dimensional subspace of L p , p >
2, then S n,K ≤ C √ p with some absolute constant C ; see [15]. These resultsare implied by the following estimate proved in [14]: Theorem 1.3. ([14])
For any centrally-symmetric star body K ⊆ R n and any evencontinuous non-negative function f on K , Z K f ( x ) dx ≤ d ovr ( K, I n ) max ξ ∈ S n − Z K ∩ ξ ⊥ f ( x ) dx | K | /n , where d ovr ( K, I n ) is the outer volume ratio distance from K to the class I n ofintersection bodies in R n . The class of intersection bodies I n was introduced by Lutwak [17]; it can bedefined as the closure in the radial metric of radial sums of ellipsoids centered atthe origin in R n .On the other hand, it was shown in [9] that in general the constants S n are ofthe order √ n , up to a doubly-logarithmic term. SERGEY BOBKOV, BO’AZ KLARTAG, AND ALEXANDER KOLDOBSKY
Theorem 1.4. ([9])
For any n ≥ , there exists a centrally-symmetric convex body T ⊆ R n and an even, continuous probability density f : T → [0 , ∞ ) such that, forany affine hyperplane H ⊆ R n , (1.6) Z T ∩ H f ( x ) dx ≤ C √ log log n √ n | T | − /n , where C > is a universal constant. The connection between (1.2) and the slicing inequality for arbitrary measures(1.5) is as follows.
Lemma 1.5.
Given a Borel measurable function f ≥ on R n , for any ξ ∈ S n − and p > , p ( p + 1) (cid:18) sup s ∈ R Z ( x,ξ )= s f ( x ) dx (cid:19) p Z | ( x, ξ ) | p f ( x ) dx ≥ (cid:18) Z f ( x ) dx (cid:19) p +1 . If f is defined on a set K in R n , we then have2 p ( p + 1) (cid:18) sup s ∈ R Z K ∩{ ( x,ξ )= s } f ( x ) dx (cid:19) p M K,f,p ( ξ ) ≥ (cid:18) Z K f ( x ) dx (cid:19) p +1 . The lower bound in Theorem 1.1 thus follows, by combining the above inequalitywith (1.2) and Theorem 1.4.
Corollary 1.6.
With some positive absolute constants c and C , for every p ≥ , c √ n √ log log n ≤ S n ≤ Cγ ( p, n ) . Lemma 1.5, in conjunction with Theorem 1.2, leads to a new slicing inequality.In the case of volume, where f ≡
1, this inequality was established earlier by Ball[1] for p = 1 and by Milman [18] for arbitrary p . Theorem 1.7.
Let f ≥ be a Borel measurable function on a compact set K ⊆ R n of positive volume. Then, for any p > , Z K f ( x ) dx ≤ C √ p d ovr ( K, L np ) | K | /n sup H Z K ∩ H f ( x ) dx, where the supremum is taken over all affine hyperplanes H in R n , and C is anabsolute constant. Theorem 1.7 also holds for 1 ≤ p ≤
2, but in this case it is weaker than Theorem1.3, because the unit ball of every finite dimensional subspace of L p , < p ≤ , is an intersection body; see [10]. However, for p > L p are not necessarily intersection bodies. For example the unit balls of ℓ np are notintersection bodies if p > n ≥
5; see [11, Th. 4.13]. So the result of Theorem 1.7is new for p >
2, and generalizes the estimate from [15] in the case where K itselfbelongs to the class L np .Theorem 1.7 gives another reason to estimate the outer volume ratio distance d ovr ( K, L np ) from an arbitrary symmetric convex body to the class of unit balls ofsubspaces of L p . As mentioned before, d ovr ( K, L np ) ≤ √ n, STIMATES FOR MOMENTS OF GENERAL MEASURES ON CONVEX BODIES 5 uniformly over all centrally-symmetric convex bodies K in R n . Surprisingly, thecorresponding lower estimates seem to be missing in the literature. CombiningTheorems 1.7 and 1.4, we get a lower estimate which shows that √ n is optimal upto a doubly-logarithmic term with respect to the dimension n and a term dependingon the power p only. Corollary 1.8.
There exists a centrally-symmetric convex body T ⊆ R n such that d ovr ( T, L np ) ≥ c √ n √ p log log n for every p ≥ , where c > is a universal constant. We end the Introduction with a comparison result for the quantities M K,f,p ( ξ ).For p ≥
1, introduce the Banach-Mazur distance d BM ( M, L np ) = inf (cid:8) a ≥ ∃ D ∈ L np such that D ⊂ M ⊂ aD (cid:9) from a star body M in R n to the class L np . Recall that L np is invariant with respectto linear transformations. By John’s theorem, if M is origin-symmetric and convex,then d BM ( M, L np ) ≤ √ n . We prove the following: Theorem 1.9.
Let K and M be origin-symmetric star bodies in R n , and let f ≥ be an even continuous function on R n . Given p ≥ , suppose that for every ξ ∈ S n − (1.7) Z K | ( x, ξ ) | p f ( x ) dx ≤ Z M | ( x, ξ ) | p f ( x ) dx. Then Z K f ( x ) dx ≤ d pBM ( M, L np ) Z M f ( x ) dx. This result is in the spirit of the isomorphic Busemann-Petty problem for arbi-trary measures proved in [16]: with the same notations, if Z K ∩ ξ ⊥ f ( x ) dx ≤ Z M ∩ ξ ⊥ f ( x ) dx, ∀ ξ ∈ S n − , then Z K f ( x ) dx ≤ d BM ( K, I n ) Z M f ( x ) dx. We refer the reader to [11, Ch.5] for more about the Busemann-Petty problem.Throughout this paper, we write a ∼ b when ca ≤ b ≤ Ca for some absoluteconstants c, C. A convex body K in R n is a compact, convex set with a non-emptyinterior. The standard scalar product between x, y ∈ R n is denoted by ( x, y ) andthe Euclidean norm of x ∈ R n by | x | . We write log for the natural logarithm.2. Proofs
In this section we prove Theorem 1.2, Lemma 1.5 and Theorem 1.9. The otherresults of this paper will follow as explained in the Introduction.Given a compact set K ⊆ R n and x ∈ R n , put k x k K = min { a ≥ x ∈ aK } , if x ∈ aK for some a ≥
0, and k x k K = ∞ in the other case. For star bodies, itrepresents the usual Minkowski functional associated with K . SERGEY BOBKOV, BO’AZ KLARTAG, AND ALEXANDER KOLDOBSKY
Proof of Theorem 1.2.
Let D ⊆ R n be the unit ball of an n -dimensional subspaceof L p , so that the relation (1.3) holds for some measure ν D on the unit sphere S n − .Then, integrating the inequalitymin θ ∈ S n − Z K | ( x, θ ) | p dµ ( x ) ≤ Z K | ( x, ξ ) | p dµ ( x ) ( ξ ∈ S n − )over the variable ξ with respect to ν D , we get the relation ν D ( S n − ) min θ ∈ S n − Z K | ( x, θ ) | p dµ ( x ) ≤ Z K k x k pD dµ ( x ) . In the case K ⊆ D , we have k x k D ≤ k x k K ≤ K , so that the last integral doesnot exceed µ ( K ) = 1, and thus(2.1) ν D ( S n − ) min θ ∈ S n − Z K | ( x, θ ) | p dµ ( x ) ≤ . In order to estimate the left-hand side of (2.1) from below, we represent the value ν D ( S n − ) as the integral R S n − | x | p dν D ( x ) and apply the well-known formula | x | p = Γ( p + n )2 π n − Γ( p +12 ) Z S n − | ( x, θ ) | p dθ, x ∈ R n (see for example [11, Lemma 3.12]). Using (1.3), this yields the representation ν D ( S n − ) = Γ( p + n )2 π n − Γ( p +12 ) Z S n − Z S n − | ( x, θ ) | p dθ dν D ( x )= Γ( p + n )2 π n − Γ( p +12 ) Z S n − k θ k pD dθ. The last integral may be related to the volume of D , by using the polar formulafor the volume of D , n | D | = Z S n − k θ k − nD dθ = s n − Z S n − k θ k − nD dσ n − ( θ ) , where σ n − denotes the normalized Lebesgue measure on S n − and s n − = π n Γ( n ) is its ( n − Z k θ k − nD dσ n − ( θ ) ≥ (cid:18) Z k θ k pD dσ n − ( θ ) (cid:19) − np , or equivalently Z k θ k pD dθ ≥ s p + nn n − ( n | D | ) − pn . Thus, ν D ( S n − ) ≥ Γ( p + n ) s p + nn n − π n − Γ( p +12 ) n pn | D | pn = √ π Γ( p + n )Γ( p +12 ) Γ( n ) (cid:16) s n − n | D | (cid:17) pn ≥ c p Γ( p +12 ) | D | pn , where c > √ n s n n − → c as n → ∞ , for some absolute c >
0, as well as the estimateΓ( p + n ) / Γ( n ) ≥ ( cn ) p/ . STIMATES FOR MOMENTS OF GENERAL MEASURES ON CONVEX BODIES 7
Applying this lower estimate on the left-hand side of (2.1), we getmin θ ∈ S n − Z K | ( x, θ ) | p dµ ( x ) ≤ C p Γ (cid:16) p + 12 (cid:17) | D | pn . It remains to take the minimum over all admissible D and note that Γ (cid:0) p +12 (cid:1) /p ≤ c √ p for p ≥ (cid:3) To prove Lemma 1.5, we need the following simple assertion.
Lemma 2.1.
Given a measurable function g : R → [0 , , the function q (cid:18) q + 12 Z ∞−∞ | t | q g ( t ) dt (cid:19) q +1 is non-decreasing on ( − , ∞ ) . Proof.
The standard argument is similar to the one used in the proof of Lemma2.4 in [7]. Given − < q < p , let A > Z ∞−∞ | t | q g ( t ) dt = Z A − A | t | q dt = 2 q + 1 A q +1 . Using | t | p ≤ A p − q | t | q ( | t | ≤ A ) and | t | p ≥ A p − q | t | q ( | t | ≥ A ) , together with the assumption 0 ≤ g ≤
1, we then have Z | t |≤ A (1 − g ( t )) | t | p dt − Z | t | >A g ( t ) | t | p dt ≤ A p − q (cid:16) R | t |≤ A (1 − g ( t )) | t | q dt − R | t | >A g ( t ) | t | q dt (cid:17) = 0 . Hence Z ∞−∞ g ( t ) | t | p dt ≥ Z A − A | t | p dt = 2 p + 1 A p +1 , that is, (cid:18) p + 12 Z ∞−∞ g ( t ) | t | p dt (cid:19) p +1 ≥ A = (cid:18) q + 12 Z ∞−∞ g ( t ) | t | q dt (cid:19) q +1 . (cid:3) Proof of Lemma 1.5.
One may assume that f is integrable. For t ∈ R , introducethe hyperplanes H t = { ( x, ξ ) = t } . Since f is Borel measurable on R n , the function g ( t ) = R H t f ( x ) dx sup s R H s f ( x ) dx is Borel measurable on the line and satisfies k g k ∞ = 1. By Fubini’s theorem, Z ∞−∞ | t | p g ( t ) dt = R | ( x, ξ ) | p f ( x ) dx sup s R H s f ( x ) dx , Z ∞−∞ g ( t ) dt = R f ( x ) dx sup s R H s f ( x ) dx . SERGEY BOBKOV, BO’AZ KLARTAG, AND ALEXANDER KOLDOBSKY
Applying Lemma 2.1 to the function g with q = 0 and p , we get12 Z ∞−∞ g ( t ) dt ≤ (cid:18) p + 12 Z ∞−∞ | t | p g ( t ) dt (cid:19) p +1 , which in our case becomes (cid:16) Z f ( x ) dx (cid:17) p +1 ≤ ( p + 1) (cid:16) s Z H s f ( x ) dx (cid:17) p Z | ( x, ξ ) | p f ( x ) dx. (cid:3) Proof of Theorem 1.9.
Let D ∈ L np be such that the distance d ovr ( M, L np ) isalmost realized, i.e., for small δ >
0, suppose that D ⊆ M ⊆ (1 + δ ) d BM ( M, L np ) D .Integrating both sides of (1.7) over ξ ∈ S n − with respect to the measure ν D from (1.3), we get Z K k x k pD f ( x ) dx ≤ Z M k x k pD f ( x ) dx. Equivalently, using the integrals in spherical coordinates, we have0 ≤ Z S n − k θ k pD (cid:18) Z k θ k − M k θ k − K r n + p − f ( rθ ) dr (cid:19) dθ = Z S n − k θ k pD k θ k pM I ( θ ) dθ, where I ( θ ) = k θ k pM Z k θ k − M k θ k − K r n + p − f ( rθ ) dr. For θ ∈ S n − such that k θ k K ≥ k θ k M , the latter quantity is non-negative, and onemay proceed by writing I ( θ ) = Z k θ k − M k θ k − K (cid:16) k θ k pM − r − p (cid:17) r n + p − f ( rθ ) dr + Z k θ k − M k θ k − K r n − f ( rθ ) dr ≤ Z k θ k − M k θ k − K r n − f ( rθ ) dr. But, in the case k θ k K ≤ k θ k M , we have − I ( θ ) = k θ k pM Z k θ k − K k θ k − M r p r n − f ( rθ ) dr ≥ Z k θ k − K k θ k − M r n − f ( rθ ) dr, which is the same upper bound on I ( θ ) as before. Thus,0 ≤ Z S n − k θ k pD k θ k pM (cid:18) Z k θ k − M k θ k − K r n − f ( rθ ) dr (cid:19) dθ, that is, Z S n − k θ k pD k θ k pM (cid:18) Z k θ k − K r n − f ( rθ ) dr (cid:19) dθ ≤ Z S n − k θ k pD k θ k pM (cid:18) Z k θ k − M r n − f ( rθ ) dr (cid:19) dθ. Now, by the choice of D , k θ k M ≤ k θ k D ≤ (1 + δ ) d BM ( M, L np ) k θ k M STIMATES FOR MOMENTS OF GENERAL MEASURES ON CONVEX BODIES 9 for every θ ∈ S n − . Hence Z K f ( x ) dx = Z S n − (cid:18) Z k θ k − K r n − f ( rθ ) dr (cid:19) dθ ≤ Z S n − k θ k pD k θ k pM (cid:18) Z k θ k − K r n − f ( rθ ) dr (cid:19) dθ ≤ Z S n − k θ k pD k θ k pM (cid:18) Z k θ k − M r n − f ( rθ ) dr (cid:19) dθ ≤ (1 + δ ) d pBM ( M, L np ) Z S n − (cid:18) Z k θ k − M r n − f ( rθ ) dr (cid:19) dθ = (1 + δ ) d pBM ( M, L np ) Z M f ( x ) dx. Sending δ to zero, we get the result. (cid:3) References [1] K. Ball,
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Department of Mathematics, University of Minnesota, 206 Church St SE, Minneapo-lis, MN 55455 USA
E-mail address : [email protected] Department of Mathematics, Weizmann Institute of Science, Rehovot 76100 Israel,and School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978
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