Interpolating Thin-Shell and Sharp Large-Deviation Estimates For Isotropic Log-Concave Measures
aa r X i v : . [ m a t h . F A ] J un Interpolating Thin-Shell and Sharp Large-DeviationEstimates For Isotropic Log-Concave Measures
Olivier Gu´edon and Emanuel Milman Abstract
Given an isotropic random vector X with log-concave density in Euclidean space R n , we study the concentration properties of | X | on all scales, both above and belowits expectation. We show in particular that: P ( (cid:12)(cid:12) | X | − √ n (cid:12)(cid:12) ≥ t √ n ) ≤ C exp( − cn min( t , t )) ∀ t ≥ , for some universal constants c, C >
0. This improves the best known deviation re-sults on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively,and recovers the sharp large-deviation estimate of Paouris. Another new feature ofour estimate is that it improves when X is ψ α ( α ∈ (1 , p V ar( | X | ) weobtain is of the order of n / , and improves down to n / when X is ψ . Our esti-mates thus continuously interpolate between a new best known thin-shell estimateand the sharp large-deviation estimate of Paouris. As a consequence, a new bestknown bound on the Cheeger isoperimetric constant appearing in a conjecture ofKannan–Lov´asz–Simonovits is deduced. Let a Euclidean norm |·| on R n be fixed. This work is dedicated to quantitative concen-tration properties of | X | , where X is an isotropic random vector in R n with log-concavedensity. Recall that a random vector X in R n (and its density) is called isotropic if E X = 0 and E X ⊗ X = Id , i.e. its barycenter is at the origin and its covariance matrixis equal to the identity one. For such an X , if A ∈ M n ( R ) denotes an n by n matrix, ob-serve that E | AX | = k A k HS , where k A k HS = qP i,j A i,j denotes the Hilbert–Schmidtnorm of A . Here and throughout we use E to denote expectation, P to denote prob-ability, and V ar to denote variance. A function g : R n → R + is called log-concave if Universit´e Paris-Est Marne La Vall´ee, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees. 5,Bd Descartes, Champs sur Marne 77454, Marne La Vall´ee, C´edex 2, France. Email: [email protected]. Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel. Supportedby ISF and the Taub Foundation (Landau Fellow). Email: [email protected]. log g : R n → R ∪ { + ∞} is convex. Throughout this work, C , c , c , C ′ , etc. denote uni-versal positive numeric constants, independent of any other parameter and in particularthe dimension n , whose value may change from one occurrence to the next.It was conjectured by Anttila, Ball and Perissinaki [1] that | X | is concentrated aroundits expectation significantly more than suggested by the trivial bound V ar | X | ≤ E | X | = n . Namely, they conjectured that there exists a sequence { ε n } decreasing to 0 with thedimension n , so that X is concentrated within a “thin shell” of relative width 2 ε n aroundthe (approximately) expected Euclidean norm of √ n : P ( (cid:12)(cid:12) | X | − √ n (cid:12)(cid:12) ≥ ε n √ n ) ≤ ε n . (1.1)Their conjecture was mainly motivated by the Central Limit Problem for log-concavemeasures, and as pointed out in [1], implies that most marginals of log-concave measuresare approximately Gaussian.A stronger version of this conjecture was put forth by Bobkov and Koldobsky [9].It may be equivalently formulated as stating that the “thin-shell width” p V ar | X | isbounded above by a universal constant C .An even stronger conjecture is due to Kannan, Lov´asz and Simonovits [21]. In anequivalent form, it states that for any smooth function f : R n → R : V ar( f ( X )) ≤ C E |∇ f ( X ) | . Applied to the function f ( x ) = | x | p with p = c √ n , the KLS conjecture implies (see [14]and Section 4) that: P ( (cid:12)(cid:12) | X | − √ n (cid:12)(cid:12) ≥ t √ n ) ≤ C exp( − c √ nt ) ∀ t ≥ . (1.2)It was shown by G. Paouris [34] that the predicted positive deviation estimate (1.2)indeed holds in the large: P ( | X | ≥ (1 + t ) √ n ) ≤ exp( − c √ nt ) ∀ t ≥ C > . (1.3)Moreover, Paouris showed that when A ∈ M n ( R ) with k A k HS = n , and X is ψ α ( α ∈ [1 , b α >
0, then: P ( | AX | ≥ (1 + t ) √ n ) ≤ exp( − c ( n/ ( b α k A k op )) α t ) ∀ t ≥ C > . (1.4)Here k A k op denotes the operator norm of A . Recall that X (and its density) is said tobe “ ψ α with constant b α ” if:( E |h X, y i| p ) /p ≤ b α p /α (cid:16) E |h X, y i| (cid:17) / ∀ p ≥ ∀ y ∈ R n . Note that this definition is linearly invariant and that necessarily b α ≥ − /α . We willsimply say that “ X is ψ α ”, if it is ψ α with a universal positive constant C . By a resultof Berwald [5] or by Borell’s Lemma [12] (see [32, Appendix III]), it is well known that2ny X with log-concave density is ψ with b ≤ C , some universal constant, and so weonly gain additional information when α > P ( | AX | ≤ ε √ n ) ≤ ( Cε ) c ( n/ ( b α k A k op )) α ∀ ε ∈ (0 , /C ) , (1.5)for some constant C > α ∈ [1 , P ( | X | ≥ (1 + t ) √ n ) when t ∈ [0 , C ], and negative deviation P ( | X | ≤ (1 − t ) √ n ) when t ∈ [0 , c ] ( c ∈ (0 , p V ar | X | .In a breakthrough work, the first non-trivial estimate on the concentration of | X | around its expectation was given by B. Klartag in [23], involving delicate logarithmicimprovements in n over the trivial bounds. This validated the conjectured thin-shell con-centration (1.1), allowing Klartag to resolve the Central Limit Problem for log-concavemeasures. A different proof continuing Paouris’ approach was given by Fleury, Gu´edonand Paouris in [16]. Klartag then improved in [24] his estimates from logarithmic topolynomial in n as follows (for any small ε > P ( (cid:12)(cid:12) | X | − √ n (cid:12)(cid:12) ≥ t √ n ) ≤ C ε exp( − c ε n − ε t − ε ) ∀ t ∈ [0 , . (1.6)This implies in particular a thin-shell estimate of: p V ar | X | ≤ C ε n − + ε . Note, however, that when t = 1 /
2, (1.6) does not recover the sharp positive large-deviation estimate of Paouris (1.3).Recently in [15], B. Fleury improved Klartag’s thin-shell estimate to: p V ar | X | ≤ Cn − , by obtaining the following deviation estimates: P ( | X | ≥ (1 + t ) √ n ) ≤ C exp( − cn t ) ∀ t ∈ [0 ,
1] ; P ( | X | ≤ (1 − t ) √ n ) ≤ C exp( − cn t ) ∀ t ∈ [0 , . Note, however, that when t = 1 /
2, Fleury’s positive and negative large-deviation es-timates are both inferior to those of Klartag, and so in the mesoscopic scale t = n − δ ( δ > ψ α condition, contrary to the ones of Paouris. See also [10, 25, 14, 28]for further related results.All of this suggests that one might hope for a concentration estimate which:3 Recovers Paouris’ sharp positive large-deviation estimate (1.4). • Improves if X is ψ α . • Improves the best-known thin-shell estimate of Fleury. • Improves the best-known mesoscopic-deviation estimate of Klartag. • Interpolates continuously between all scales of t (bulk, mesoscopic, large-deviation).The aim of this work is to provide precisely such an estimate. Following Paouris, we formulate our main results in greater generality, allowing an ap-plication of a linear transformation to X . Theorem 1.1.
Let X denote an isotropic random vector in R n with log-concave density,which is in addition ψ α ( α ∈ [1 , ) with constant b α , and let A ∈ M n ( R ) satisfy k A k HS = n . Then: P ( (cid:12)(cid:12) | AX | − √ n (cid:12)(cid:12) ≥ t √ n ) ≤ C exp( − cη α min( t α , t )) ∀ t ≥ , (1.7) where: η := n k A k op b α . (1.8) In particular, we obtain the following thin-shell estimate: p V ar ( | AX | ) ≤ Cn η − α α ) . (1.9)For concreteness and future reference, we state again the deviation estimates aboveand below the expectation separately: the constant C in (1.7) may actually be removedin the former estimate: P ( | AX | ≥ (1 + t ) √ n ) ≤ exp( − cη α min( t α , t )) ∀ t ≥ P ( | AX | ≤ (1 − t ) √ n ) ≤ C exp( − cη α max( t α , log c − t )) ∀ t ∈ [0 , . (1.11)Applying Theorem 1.1 with α = 1 and A = Id , we obtain that for any isotropic X with log-concave density, the above estimates hold with η ≥ cn , and in particular wededuce the following improved thin-shell estimate: p V ar( | X | ) ≤ Cn − . (1.12)4lso note that (1.10) recovers (up to constants) Paouris’ sharp large-deviation estimate(1.4). Moreover, we obtain P ( | AX | ≥ (1 + t ) √ n ) ≤ exp( − C t η α ) and P ( | AX | ≤ ε √ n ) ≤ C ′ exp( − C ε η α ) for any t > ε ∈ (0 , t ≥ C and ε ∈ (0 , /C ), for some large enough C >
0. Itis also possible to recover Paouris’ small-ball estimate (1.5), but this seems to requireadditional justification, which we leave for another note.Theorem 1.1 is a standard consequence of (and essentially equivalent to) the followingmoment estimates, which are the main result of this work:
Theorem 1.2.
With the same assumptions and notation as in Theorem 1.1, for any ≤ | p − | ≤ c η α α +2) : − C | p − | η αα +2 ≤ ( E | AX | p ) p ( E | AX | ) ≤ C | p − | η αα +2 , (1.13) and for any c η α α +2) ≤ | p − | ≤ c η α : − C (cid:18) | p − | η α (cid:19) α +1 ≤ ( E | AX | p ) p ( E | AX | ) ≤ C (cid:18) | p − | η α (cid:19) α +1 . (1.14)More precisely, we first derive a refined version of Theorem 1.2 with AX replacedby Y = ( AX + G n ) / √
2, where G n denotes an independent standard Gaussian randomvector in R n . From this version, we derive the deviation estimates (1.10) and (1.11) for Y directly. Theorem 1.1 for AX then easily follows, but to deduce back the negative moment estimates in (1.14) for AX up to − p = c η α (or equivalently, the negative deviation estimate (1.11)), we elude to the small-ball estimate (1.5). We remark thatthe lower bound | p − | ≥ C > L -normis interpreted as exp( E log | AX | ). Remark 1.3.
Our choice to present the results assuming that k A k HS = n is purely foraesthetic reasons, facilitating the comparison to the previously known results. Indeed,we can obviously remove this assumption by scaling X , and state all of our deviationestimates around (and relative to) the expected value ( E | AX | ) / = k A k HS instead of √ n . This leads to the following scale-invariant definition of η as η := k A k HS / ( b α k A k op ),which naturally also appears in the work of Paouris [34, 35].Let us finally mention that by a standard application of a remarkable theorem due toBobkov [6], we improve the best-known general bound on the Cheeger constant D Che ( µ )of a probability measure µ in R n with isotropic log-concave density (we refer to [6, 29] formissing definitions and background). Bobkov’s theorem states that for such measures D Che ( µ ) ≥ c/ ( E | X | p V ar | X | ) (where X is distributed according to µ ), and so ourimproved thin-shell estimate (1.12) implies:5 orollary 1.4. Let µ denote a probability measure in R n with isotropic log-concavedensity. Then D Che ( µ ) ≥ cn − . This should be compared to the bound D Che ( µ ) ≥ c > D Che ( µ ) ≥ cn − when the density of µ is ψ . We assume throughout all proofs in this work that η , and hence n , are greater than somelarge enough positive constant, since otherwise all stated results follow trivially (or easily,by inspecting the proof). Let G n,k denote the Grassmann manifold of all k -dimensionallinear subspaces of R n , and SO ( n ) the group of rotations. Fixing a Euclidean structureon R n , and given a linear subspace F , we denote by S ( F ) and B ( F ) the unit-sphere andunit-ball in F , respectively. When F = R n , we simply write S n − and B n . We denote by P F the orthogonal projection onto F in R n , and given a random vector Y with density g , we denote by π F g the marginal density of g on F , i.e. the density of P F Y . When F = span( θ ), θ ∈ S n − , we denote by π θ g the density on R given by π θ g ( t ) := π F g ( tθ ).For the proof of Theorem 1.2, we use many of the ingredients developed previouslyby Klartag [24], and adapted to the language of moments by Fleury [14, 15]: • It is (almost) enough to verify (1.13) and (1.14) with AX replaced by Y = ( AX + G n ) / √ • It is useful to first project Y onto a lower-dimensional subspace F ∈ G n,k . Thisidea also appears in essence in the work of Paouris [34]. Klartag and Paourisuse V. Milman’s approach to Dvoretzky’s theorem [30, 32] for identifying lower-dimensional structures in most marginals P F Y . Fleury, on the other hand, takes anaverage over the Haar measure on G n,k , which is more efficient (see [15] or below):( E | Y | p ) /p ( E | Y | ) / ≤ ( E F,Y | P F Y | p ) /p ( E F,Y | P F Y | ) / . (1.15) • Rewriting using the invariance of the Haar measure and polar coordinates:( E F,Y | P F Y | p ) /p ( E F,Y | P F Y | ) / = ( E U h k,p ( U )) /p ( E U h k, ( U )) / , (1.16)where U is uniformly distributed over SO ( n ), E ∈ G n,k , θ ∈ S ( E ), g denotesthe density of Y in R n , and h k,p : SO ( n ) → R + is defined as: h k,p ( u ) := Vol( S k − ) Z ∞ t p + k − π u ( E ) g ( tu ( θ )) dt . (1.17)To control the ratio in (1.16), a good bound on the log-Lipschitz constant L k,p of h k,p is required. 6ur main technical result in this work is the following improvement over the log-Lipschitz bounds of Klartag from [24]: Theorem 1.5.
Under the same assumptions as in Theorem 1.1, if p ≥ − k + 1 then L k,p ≤ C k A k op b α max( k, p ) /α +1 / . Contrary to Klartag’s analytical approach for controlling the log-Lipschitz constant, oursis completely based on geometric convexity arguments, employing the convex bodies K k + q introduced by K. Ball in [3], and a variation on the L q -centroid bodies, which wereintroduced by E. Lutwak and G. Zhang in [27].Fleury proceeds by employing three additional ingredients: • As shown by Borell [11] (see also [4]), for any log-concave density w on R + : q log R ∞ t q w ( t ) dt Γ( q + 1) is concave on R + . (1.18)Consequently, p log( h k,p ( u ) / Γ( k + p )) is concave on p ∈ [ − k + 1 , ∞ ) for anyfixed u ∈ SO ( n ). This ingredient was also used in [16]. • As follows e.g. from the work of Bakry and ´Emery [2] (see also [26]), for anyLipschitz function f : SO ( n ) → R + , the following log-Sobolev inequality is satisfied(see Sections 2 and 3 for definitions): E nt U ( f ) ≤ cn E U ( |∇ f | /f ) . (1.19) • The latter log-Sobolev inequality implies via the Herbst argument, that for anylog-Lipschitz function f : SO ( n ) → R + with log-Lipschitz constant bounded aboveby L , the following reverse H¨older inequality holds (see [15, (15)]):( E U f q ) q ≤ exp (cid:18) C L n ( q − r ) (cid:19) ( E U f r ) r ∀ q > r > . (1.20)We proceed by using these ingredients as our predecessors, but our proof correctsthe slight inefficiency of Fleury’s approach in the resulting large-deviation estimate (wit-nessed by the comparison to Klartag’s estimate earlier). The improvement here comesfrom the fact that we take the derivative in p of (1.15), and optimize on the dimension k for each p separately, as opposed to optimizing on a single k directly in (1.15). However,this by itself would not yield the improvement in the thin-shell estimate - the latter isdue to our improved log-Lipschitz estimate in Theorem 1.5. Only by combining thisimproved log-Lipschitz estimate with our variation on Fleury’s method, are we able torecover the sharp large-deviation estimates of Paouris (1.4). Moreover, the negative mo-ment estimates of (1.13) and (1.14) are also obtained almost for free, at least with AX replaced by Y , after some slight additional justification for handling the p moments inthe range p ∈ [ − cη − / , cη − / ]. 7he rest of this work is organized as follows. In Section 2 we prove a more generalversion of Theorem 1.5. In Section 3 we provide a complete proof of a refined version ofTheorem 1.2, with AX replaced by Y , without eluding to (1.5). In Section 4, we derivefor completeness Theorem 1.1 from Theorem 1.2, and obtain the reduction from AX to Y . In the Appendix, we provide a proof of Proposition 2.6 and other lemmas, whosepurpose is to handle the case when X is not centrally-symmetric (has non-even density). Acknowledgement.
We thank Bo’az Klartag for his interest and comments andMatthieu Fradelizi for discussions. We also thank the anonymous referees for helpfulsuggestions. This work was done in part when the authors attended the Thematic Pro-gram on Asymptotic Geometric Analysis at the Fields Institute in Toronto.
Let M k,l ( R ) denote the set of k by l matrices over R , and set M n ( R ) = M n,n ( R ). Weequip SO ( n ) = (cid:8) u ∈ M n ( R ); u t u = Id , det ( u ) = 1 (cid:9) with its standard (left and right) invariant Riemannian metric g , which we specify forconcreteness on T Id SO ( n ), the tangent space at the identity element Id ∈ SO ( n ).Fixing an orthonormal basis of R n and taking the derivative of the relation u t u = Id , we see that this tangent space may be identified with all anti-symmetric matrices (cid:8) B ∈ M n ( R ); B t + B = 0 (cid:9) . Given B ∈ T Id SO ( n ), we set | B | = h B, B i := g Id ( B, B ) = k B k HS , where recall the Hilbert-Schmidt norm of A ∈ M k,l ( R ) is given by k A k HS := tr ( A t A ) = P ≤ i ≤ k, ≤ j ≤ l A i,j . The factor of above is simply a convenience to ensurethat a full 2 π degree rotation in any two-plane leaving the orthogonal complement inplace, has geodesic length 2 π , and to prevent further appearances of factors like √ SO ( n ) ⊂ M n ( R ) when M n ( R ) is equipped with the Hilbert-Schmidt metric(i.e. identified with the canonical Euclidean space on its n entries). Throughout this section, let Y denote a random vector in R n with log-concave density g and barycenter at the origin. Given an integer k between 1 and n , a real number p ≥ − k + 1, a linear subspace E ∈ G n,k and θ ∈ S ( E ), we recall the definition of thefunction h k,p : SO ( n ) → R + : h k,p ( u ) := Vol( S k − ) Z ∞ t p + k − π u ( E ) g ( tu ( θ )) dt , u ∈ SO ( n ) . (2.1)Note that π E g is log-concave for any E ∈ G n,k by the Pr´ekopa–Leindler Theorem (e.g.[18]). 8hen Y = ( X + G n ) / √
2, where (as throughout this work) X denotes an isotropicrandom vector in R n with log-concave density, an upper bound on the log-Lipschitzconstant (i.e. the Lipschitz constant of the logarithm) of: SO ( n ) ∋ u π u ( E ) g ( tu ( θ ))was obtained by Klartag [24, Lemma 3.1], playing a crucial role in his polynomial esti-mates on the thin-shell of an isotropic log-concave measure. When t ≤ C √ k , Klartag’sestimate is of the order of k . In [15], Fleury defined a truncated version of (2.1), wherethe integral ranges up to C √ k . Klartag’s estimate obviously implies the same bound onthe log-Lipschitz constant of this truncated version of h k,p .Our main technical result in this work is the following improved estimate on the log-Lipschitz constant of h k,p , which is completely based on geometric convexity arguments.Note that we do not need any truncation, nor do we need to assume that Y has beenconvolved with a Gaussian to obtain a meaningful estimate. However, the improvementover Klartag’s k bound appears after this convolution. Theorem 2.1.
The log-Lipschitz constant L k,p of h k,p ( u ) : SO ( n ) → R + is boundedabove by C max( k, p ) dist ( Z +max( k,p ) ( g ) , B n ) . Here Z + q ( w ) ⊂ R n ( q ≥
1) denotes the one-sided L q -centroid body of the density w (which may not have total mass one), defined via its support functional: h Z + q ( w ) ( y ) = (cid:18) Z R n h x, y i q + w ( x ) dx (cid:19) /q , (here as usual a + := max( a, q ∈ (0 , w is even, this coincides with the more standarddefinition of the L q -centroid body, introduced by E. Lutwak and G. Zhang in [27] (undera different normalization): h Z q ( w ) ( y ) = (cid:18)Z R n |h x, y i| q w ( x ) dx (cid:19) /q . Clearly: Z + q ( w ) ⊂ /q Z q ( w ) . In any case, when w is the characteristic function of a set K , we denote Z + q ( K ) := Z + q (1 K ), and similarly for Z q ( K ). Lastly, the geometric distance dist( K, L ) between twosubsets
K, L ⊂ R n is defined as:dist( K, L ) := inf { C /C ; C L ⊂ K ⊂ C L , C , C > } . A very useful result for handling the non-even case is due to Gr¨unbaum [19] (see also[17, Formula (10)] or [7, Lemma 3.3] for simplified proofs):9 emma 2.2 (Gr¨unbaum) . Let X denote a random variable on R with log-concavedensity and barycenter at the origin. Then e ≤ P ( X ≥ ≤ − e . Note that by definition, Y (and its density g ) is ψ α ( α ∈ [1 , b α iff Z q ( g ) ⊂ b α q /α Z ( g ) for all q ≥
2. Also recall that by a result of Berwald [5] or as aconsequence of Borell’s Lemma [12] (see also [31] or [32, Appendix III]), a log-concaveprobability density g is always ψ , and that moreover:1 ≤ q ≤ q ⇒ Z q ( g ) ⊂ Z q ( g ) ⊂ C q q Z q ( g ) . (2.2)If in addition the barycenter of g is at the origin, then repeating the argument leadingto (2.2) and using Lemma 2.2, one verifies:1 ≤ q ≤ q ⇒ (cid:18) e (cid:19) q − q Z + q ( g ) ⊂ Z + q ( g ) ⊂ C (cid:18) e − e (cid:19) q − q q q Z + q ( g ) . (2.3)When g is isotropic, note that Z ( g ) = B n , and one may similarly show (see LemmaA.4) that cB n ⊂ Z +2 ( g ) ⊂ √ B n . It follows immediately from (2.3) that in that casedist( Z + k ( g ) , B n ) ≤ Ck , and we see that Theorem 2.1 recovers Klartag’s k order ofmagnitude when p ≤ k (which is the case of interest in the subsequent analysis).The improvement over Klartag’s bound comes from the following elementary: Lemma 2.3.
Let X denote an isotropic random-vector in R n with log-concave density.Given A ∈ M n ( R ) , set Y = ( AX + G n ) / √ and denote by g its density. Then for all q ≥ :1. Z + q ( g ) ⊃ c √ qB n .2. If X is ψ α ( α ∈ [1 , ) with constant b α , then Z + q ( g ) ⊂ ( C k A k op b α q /α + C √ q ) B n .Proof. Given θ ∈ S n − , denote Y = π θ Y , X = π θ AX and G = π θ G n (a one-dimensional standard Gaussian random variable). We have: h qZ + q ( g ) ( θ ) = 2 E ( Y ) q + = 22 q/ E ( X + G ) q + ≥ q/ E ( G ) q + P ( X ≥ . When X is centrally-symmetric then P ( X ≥
0) = 1 /
2. In the general case, since X has log-concave density on R and barycenter at the origin, Lemma 2.2 implies that P ( X ≥ ≥ /e , and hence: h qZ + q ( g ) ( θ ) ≥ e q/ E | G | q , by the symmetry of G . An elementary calculation shows that c √ q ≤ ( E | G | q ) /q ≤ c √ q for all q ≥
1, concluding the proof of the first assertion. Similarly:12 h qZ + q ( g ) ( θ ) ≤ h qZ q ( g ) ( θ ) = E | Y | q = E (cid:12)(cid:12)(cid:12)(cid:12) X + G √ (cid:12)(cid:12)(cid:12)(cid:12) q ≤ q − q/ E ( | X | q + | G | q ) . X is ψ α with constant b α and isotropic, it follows that:( E | X | q ) /q ≤ b α q /α ( E | X | ) / ≤ b α q /α k A k op , and the second assertion readily follows. Corollary 2.4.
Let X denote an isotropic random-vector in R n with log-concave density,which is in addition ψ α ( α ∈ [1 , ) with constant b α . Let A ∈ M n ( R ) , set Y = ( AX + G n ) / √ and denote by g the density of Y . Then:dist ( Z + q ( g ) , B n ) ≤ C (1 + k A k op b α q /α − / ) . Consequently, when k A k op ≥ , Theorem 2.1 implies that: L k,p ≤ C k A k op b α max( k, p ) /α +1 / . For convenience, we assume that 2 ≤ k ≤ n/
2, although it will be clear from theproof that this is immaterial. By the symmetry and transitivity of SO ( n ), and since E ∈ G n,k was arbitrary, it is enough to bound |∇ u log h k,p | at u = Id . We complete θ to an orthonormal basis (cid:8) θ , e , . . . , e k (cid:9) of E , and take (cid:8) e k +1 , . . . , e n (cid:9) to be anycompletion to an orthonormal basis of R n . In this basis, the anti-symmetric matrix M := ∇ Id log h k,p ∈ T Id SO ( n ) looks as follows: M = M M − M t , M = V − V t , M = V V , (2.4)where M ∈ M k,k ( R ), M ∈ M k,n − k ( R ), V ∈ M ,k − ( R ), V ∈ M ,n − k ( R ) and V ∈ M k − ,n − k ( R ). Indeed, the lower n − k by n − k block of M is clearly 0, since rotationsin E ⊥ , the orthogonal complement to E , leave π u ( E ) g and hence h k,p unaltered; andthe lower k − k − M is 0 since rotations which fix θ and act invariantlyon E preserve h k,p as well. Consequently |∇ Id log h k,p | = k V k HS + k V k HS + k V k HS .We will analyze the contribution of these three terms separately.Denote by T i ( i = 1 , ,
3) the subspace of T Id SO ( n ) having the form (2.4) with V j = 0 for j = i . Given B ∈ T i , we call the geodesic in SO ( n ) emanating from Id in the direction of B , i.e. R ∋ s u s := exp Id ( sB ) ∈ SO ( n ), a Type- i movement .By definition, dds u s | s =0 = B , and hence dds log h k,p ( u s ) | s =0 = h∇ Id log h k,p , B i . Clearly k V i k HS = sup = B ∈ T i h∇ Id log h k,p , B i / | B | , so our goal now is to obtain a uniform upperbound on the derivative of log h k,p induced by a Type- i movement.11o this end, we recall the following crucial fact, due to K. Ball [3, Theorem 5] in theeven case, and verified to still hold in the general one by Klartag [22, Theorem 2.2]: Theorem.
Let w denote a log-concave function on R m with < R w < ∞ and w (0) > .Given q ≥ , set: k x k = k x k K q ( w ) := (cid:18) q Z ∞ t q − w ( tx ) dt (cid:19) − q , x ∈ R m . Then for all x, y ∈ R m , ≤ k x k < ∞ , k x k = 0 iff x = 0 , k λx k = λ k x k for all λ ≥ ,and k x + y k ≤ k x k + k y k . We will thus say that k·k K q ( w ) defines a norm, even though it may fail to be even,and denote by K q ( w ) := { x ∈ R m ; k x k K q ( w ) ≤ } its associated convex compact unit-ball. Note that the constant q in front of the integral above is simply a convenientnormalization for later use. We also set k x k ˆ K q ( w ) := max( k x k K q ( w ) , k− x k K q ( w ) ), havingunit-ball ˆ K q ( w ) = K q ( w ) ∩ − K q ( w ). Note that the triangle inequality implies that: (cid:12)(cid:12)(cid:12) k x k K q ( w ) − k y k K q ( w ) (cid:12)(cid:12)(cid:12) ≤ k x − y k ˆ K q ( w ) . (2.5)Finally, note that since B m is centrally-symmetric, then C B m ⊂ K ∩ − K ⊂ K ⊂ C B m iff C B m ⊂ K ⊂ C B m , and hence: k x k ˆ K q ( w ) k y k K q ( w ) ≤ dist( K q ( w ) , B m ) | x || y | . (2.6) Let B ∈ T with | B | = 1 generate a Type-1 movement { u s } , and denote ξ = dds u s ( θ ) | s =0 ∈ T θ S ( R n ). Using henceforth the natural embedding T θ S ( R n ) ⊂ T θ R n ≃ R n , a Type-1movement ensures that u s is a rotation in the { θ , ξ } plane and that ξ lies in the or-thogonal complement to θ in E , so u s ( E ) = E . Also note since | B | = 1 that | ξ | = 1.Recalling the definition of h k,p , we conclude that for such a movement: h k,p ( u s ) = Vol( S k − ) Z ∞ t p + k − π E g ( tu s ( θ )) dt = c p,k k u s ( θ ) k − ( k + p ) K k + p ( π E g ) , where c p,k = Vol( S k − ) / ( k + p ) is totally immaterial. Consequently: |h∇ Id log h k,p , B i| = (cid:12)(cid:12)(cid:12)(cid:12) dds log h k,p ( u s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:12)(cid:12)(cid:12)(cid:12) = ( k + p ) (cid:12)(cid:12)(cid:12)(cid:12) dds log k u s ( θ ) k K k + p ( π E g ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:12)(cid:12)(cid:12)(cid:12) . Since (cid:12)(cid:12)(cid:12) dds k u s ( θ ) k K k + p ( π E g ) (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) dds u s ( θ ) (cid:13)(cid:13) ˆ K k + p ( π E g ) by the triangle-inequality (2.5), weconclude using (2.6) that: |h∇ Id log h k,p , B i| ≤ ( k + p ) k ξ k ˆ K k + p ( π E g ) k θ k K k + p ( π E g ) ≤ ( k + p )dist( K k + p ( π E g )) , B ( E )) . .2.2 Type-2 movement Let B ∈ T with | B | = 1 generate a Type-2 movement { u s } , and denote θ s := u s ( θ ) and ξ s := dds θ s ∈ T θ s S ( R n ). The Type-2 movement ensures that ξ ∈ E ⊥ and that u s is arotation in the { θ , ξ } = { θ s , ξ s } plane, and | B | = 1 ensures that | ξ | = 1. Denoting E the orthogonal complement to θ in E , it follows that u s rotates E into E s := u s ( E ) = E ⊕ span { θ s } . Consequently, u s leaves H := E ⊕ span { ξ } = E s ⊕ span { ξ s } ∈ G n,k +1 invariant, and therefore: h k,p ( u s ) = Vol( S k − ) Z ∞ Z ∞−∞ t p + k − π H g ( tθ s + rξ s ) drdt . Performing the change of variables r = vt , which is valid except at the negligible point t = 0, we obtain: h k,p ( u s ) = Vol( S k − ) Z ∞ Z ∞−∞ t p + k π H g ( t ( θ s + vξ s )) dvdt = c p,k Z ∞−∞ k θ s + vξ s k − ( k + p +1) K k + p +1 ( π H g ) dv , where c p,k = Vol( S k − ) / ( k + p + 1). Using that dds ξ s = − θ s and the triangle inequality(2.5) and (2.6) for k·k K k + p +1 ( π H g ) , we obtain: |h∇ Id log h k,p , B i| = (cid:12)(cid:12)(cid:12)(cid:12) dds log h k,p ( u s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( k + p + 1) sup v ∈ R k ξ − vθ k ˆ K k + p +1 ( π H g ) k θ + vξ k K k + p +1 ( π H g ) ≤ ( k + p + 1)dist( K k + p +1 ( π H g ) , B ( H )) sup v ∈ R | ξ − vθ || θ + vξ | = ( k + p + 1)dist( K k + p +1 ( π H g ) , B ( H )) , where we have used the fact that θ and ξ are orthogonal unit vectors in the last equality. Finally, we analyze the most important movement type, which is responsible for a sub-space of movements of dimension ( k − n − k ) (out of the dim G n,k + dim S k − = k ( n − k ) + ( k −
1) dimensional subspace of non-trivial movements).Let 0 = B ∈ T generate a Type-3 movement { u s } , and set e js := u s ( e j ) and f j := dds e js | s =0 , j = 2 , . . . , k . The Type-3 movement ensures that u s ( θ ) = θ and that all f j ∈ E ⊥ . Denote F := span { f , . . . , f k } , and note that by slightly perturbing B if necessary,we may assume that F is k − H = E ⊕ F ∈ G n, k − ,and notice that H is invariant under u s (since u s is an isometry acting as the identityon the orthogonal complement). Consequently, H = E s ⊕ F s , where E s := u s ( E ) and F s := u s ( F ), and therefore: h k,p ( u s ) = Vol( S k − ) Z ∞ Z F s t p + k − π H g ( tθ + y ) dydt . y = zt , we obtain (with c p,k = Vol( S k − ) / (2 k − p )): h k,p ( u s ) = Vol( S k − ) Z ∞ Z F s t p +2 k − π H g ( t ( θ + z )) dzdt = c p,k Z F s k θ + z k − (2 k − p ) K k − p ( π H g ) dz , which we rewrite, since u s is orthogonal, as: h k,p ( u s ) = c p,k Z F k θ + u s ( z ) k − (2 k − p ) K k − p ( π H g ) dz . As usual, the triangle inequality (2.5) for k·k K k − p ( π H g ) implies that: |h∇ Id log h k,p , B i| = (cid:12)(cid:12)(cid:12)(cid:12) dds log h k,p ( u s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 (cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 k − p ) sup z ∈ F k Bz k ˆ K k − p ( π H g ) k θ + z k K k − p ( π H g ) , and so by (2.6): |h∇ Id log h k,p , B i| (2 k − p )dist( K k − p ( π H g ) , B ( H )) ≤ sup z ∈ F | Bz || θ + z | ≤ k B k op sup z ∈ F | z | p | z | ≤ k B k HS √ | B | , where we have used that θ is perpendicular to F , and that k B k op ≤ k B k HS / √ B , as may be easily verified by using the Cauchy–Schwarzinequality. K m + p to Euclidean ball To conclude the proof of Theorem 2.1, it remains to control the geometric distance of K m + p ( π H g ) to a Euclidean ball, for H ∈ G n,m with m of the order of k . To this end, wecompare K m + p ( π H g ) to Z q ( π H g ) = P H Z q ( g ) for a suitably chosen q ≥
1. Our motivationcomes from the groundbreaking work of Paouris [34], who noted that: Z q ( π H g ) = Z q ( K m + q ( π H g )) , and using the inclusion Z q ( K ) ⊂ conv ( K ∪ − K ) for any set K of volume 1, obtained anupper bound on Vol( Z q ( π H g )) by bounding above Vol( K m + q ( π H g )), enabling Paouristo deduce important features of P H Z q ( g ). In this work, on the other hand, we take theconverse path, passing from K m + q bodies to Z q ones, and consequently need to introducethe Z + q bodies to handle non-even densities. Moreover, we require bounds on Z + q ( K )both from above and from below, which turn out to be more laborious in the non-evencase (when K is not centrally-symmetric).Since the distance to the Euclidean ball cannot increase under orthogonal projections,and since c Z + k ( g ) ⊂ c Z + m ( g ) ⊂ c Z +2 k − ( g ) ⊂ c Z + k ( g ) when k ≤ m ≤ k − heorem 2.5. Let w denote a log-concave function on R m with < R w < ∞ andbarycenter at the origin. Then for any p ≥ − m + 1 :dist ( K m + p ( w ) , B m ) ≤ C max (cid:18) mm + p , (cid:19) dist ( Z +max( p,m ) ( w ) , B m ) . For the proof, we recall several useful properties of the bodies K q ( w ) and Z + q ( K ).First, it is known (see [4, 3, 31] for the even case and [22, Lemmas 2.5,2.6] or [35, Lemma3.2 and (3.12)] for the general one) that under the assumptions of Theorem 2.5:1 ≤ q ≤ q ⇒ e − m ( q − q ) K q ( w ) w (0) /q ⊂ K q ( w ) w (0) /q ⊂ Γ( q + 1) /q Γ( q + 1) /q K q ( w ) w (0) /q . (2.7)Second, integration in polar coordinates (cf. [34]) directly shows that: Z + q ( K m + q ( w )) = Z + q ( w ) . (2.8)Lastly, we require the following proposition, which is well-known in the even-case (e.g.[33, Lemma 4.1]), but requires more work in the general one (note for instance that thebarycenter of K m + q ( w ) below need not be at the origin); its proof is postponed to theAppendix. Proposition 2.6.
For any q ≥ : C Z + q ( K m + q ( w )) ⊂ Vol ( K m + q ( w )) /q K m + q ( w ) ⊂ C Z + q ( K m + q ( w )) (cid:18) Γ( m + q + 1)Γ( m )Γ( q + 1) (cid:19) /q (2.9) Proof of Theorem 2.5.
When p ≥
1, observe that (2.9), (2.8) and Stirling’s formula,imply that: dist( K m + p ( w ) , B m ) ≤ C p + mp dist( Z + p ( w ) , B m ) , and so when p ≥ m the asserted claim follows. Otherwise, using (2.7), Stirling’s formula,(2.9) and (2.8), we see that if q ≥ max( p,
1) then:dist( K m + p ( w ) , B m ) ≤ C m + qm + p dist( K m + q ( w ) , B m ) ≤ C m + qm + p m + qq dist( Z + q ( w ) , B m ) . Setting q = m , the case p < m is also settled.The proof of Theorem 2.1 is now complete. Our goal in this section is to prove: 15 heorem 3.1.
Let X denote an isotropic random vector in R n with log-concave density,which is in addition ψ α ( α ∈ [1 , ) with constant b α . Let A ∈ M n ( R ) satisfy k A k HS = n ,and set Y := ( AX + G n ) / √ , where G n is an independent standard Gaussian randomvector in R n . Then for any | p | ≤ cη α/ : − C (cid:18) | p − | η α (cid:19) α +1 ≤ ( E | Y | p ) p ( E | Y | ) ≤ C (cid:18) | p − | η α (cid:19) α +1 . (3.1) where η was defined in (1.8). Note that by the Pr´ekopa–Leindler Theorem, Y itself has log-concave density. Wealso remark that it is possible to improve the moment estimates in the range 1 ≤ | p − | ≤ c η α α +2) exactly as in Theorem 1.2, but we do not insist on this here. SO ( n ) We start by repeating the argument of Fleury for passing from integration on R n to SO ( n ). Let 0 = | p | ≤ n − , and let k denote an integer between 2 and n to be determinedlater on, so that in addition | p | ≤ k − . Since | x | p = a n,k,p E F | P F x | p , where F is uniformlydistributed on G n,k (according to its Haar probability measure), we have: E | Y | p E | G n | p = EE F | P F Y | p EE F | P F G n | p = EE F | P F Y | p E | G k | p , where G i denotes a standard Gaussian random vector on R i . A direct calculation showsthat: E | G i | p = 2 p Γ(( p + i ) / i/ , and hence: E | Y | p = Γ(( p + n ) / k/ n/ p + k ) / EE F | P F Y | p . (3.2)Passing to polar coordinates on F ∈ G n,k and using the invariance of the Haar measureson G n,k , S ( F ) and SO ( n ) under the action of SO ( n ), we verify that: EE F | P F Y | p = E U h k,p ( U ) , (3.3)where U is uniformly distributed on SO ( n ). We now deviate from Fleury’s argument and proceed to estimate: ddp log(( E | Y | p ) p ) = ddp log(( E U h k,p ( U )) p ) + ddp (cid:18) p log Γ(( p + n ) / k/ n/ p + k ) / (cid:19) . (3.4)16iven u ∈ SO ( n ), we introduce the (non-probability) measure µ u on R + having den-sity Vol( S k − ) t k − π u ( F ) g ( tu ( θ )), where g is the density of Y on R n . We define the(probability) measure µ k,p := E U µ U on R + , and write: h k,p ( u ) = E µ u ( t p ) , E U h k,p ( U ) = E U E µ U ( t p ) = E µ k,p ( t p ) . Here and in the sequel we use the following convention: given a measure space (Ω , µ ),which does not necessarily have total mass 1, and a measurable f : Ω → R + , we set: E µ f = E µ ( f ) = Z f dµ , E nt µ ( f ) = E µ ( f log f ) − E µ ( f ) log( E µ ( f )) . A useful fact, easily verified by direct calculation, is that: ddp log(( E µ f p ) p ) = 1 p E nt µ ( f p ) E µ ( f p ) . We proceed with estimating (3.4). As explained: ddp log(( E U h k,p ( U )) p ) = 1 p E nt µ k,p ( t p ) E µ k,p ( t p ) = 1 p E nt µ k,p ( t p ) E U h k,p ( U ) . (3.5)Our main idea here is to decompose the numerator as follows: E nt µ k,p ( t p ) = E U E nt µ U ( t p ) + E nt U E µ U ( t p ) = E U E nt µ U ( t p ) + E nt U h k,p ( U ) . (3.6)The contribution of the second term in (3.6) is controlled using the log-Sobolevinequality (1.19):1 p E nt U h k,p ( U ) E U h k,p ( U ) ≤ cp n E U ( |∇ log h k,p | ( U ) h k,p ( U )) E U h k,p ( U ) ≤ cL k,p p n , (3.7)where recall L k,p denotes the log-Lipschitz constant of u h k,p ( u ). To control thecontribution of the first term in (3.6), we first write given u ∈ SO ( n ):1 p E nt µ u ( t p ) E µ u ( t p ) = ddp log(( E µ u t p ) p ) = ddp p (cid:18) log h k,p ( u )Γ( k + p ) − log h k, ( u )Γ( k ) + log Γ( k + p )Γ( k ) + log h k, ( u ) (cid:19) . By Borell’s concavity result (1.18), we realize that: ddp p (cid:18) log h k,p ( u )Γ( k + p ) − log h k, ( u )Γ( k ) (cid:19) ≤ , and hence: 1 p E nt µ u ( t p ) E µ u ( t p ) ≤ ddp (cid:18) p log Γ( k + p )Γ( k ) (cid:19) − p log h k, ( u ) . p E U E nt µ U ( t p ) E U E µ U ( t p ) ≤ ddp (cid:18) p log Γ( k + p )Γ( k ) (cid:19) + 1 p E U log(1 /h k, ( U )) h k,p ( U ) E U h k,p ( U ) . (3.8)By using the Jensen and Cauchy–Schwarz inequalities, we bound the second term by: E U log(1 /h k, ( U )) h k,p ( U ) E U h k,p ( U ) ≤ log E U h k,p ( U ) h k, ( U ) E U h k,p ( U ) ≤ log ( E U h k,p ( U ) ) / E U h k,p ( U ) ( E U h k, ( U ) − ) / ! . We now use the reverse H¨older inequality (1.20) for comparing the various momentsabove. Denoting k f k q := ( E U | f ( U ) | q ) /q , we have: k h k,p k ≤ exp CL k,p n ! k h k,p k , (cid:13)(cid:13)(cid:13) h − k, (cid:13)(cid:13)(cid:13) ≤ exp CL k, n ! (cid:13)(cid:13)(cid:13) h − k, (cid:13)(cid:13)(cid:13) = exp CL k, n ! k h k, k ≤ exp CL k, n ! k h k, k . Since k h k, k = E U h k, ( U ) = E µ k,p (1) = 1, we conclude that:1 p E U log(1 /h k, ( U )) h k,p ( U ) E U h k,p ( U ) ≤ Cp n ( L k,p + 3 L k, ) . (3.9)Now, plugging all the estimates (3.7), (3.8), (3.9) into (3.5) using the decomposition(3.6), and plugging the result into (3.4), we obtain: ddp log(( E | Y | p ) p ) ≤ cp n (2 L k,p +3 L k, )+ ddp (cid:18) p log Γ( k + p )Γ( k ) (cid:19) + ddp (cid:18) p log Γ(( p + n ) / k/ n/ p + k ) / (cid:19) . As observed by Fleury in [15], using that the function ddp log Γ( p ) is concave, one easilyverifies that the last term above satisfies: ddp (cid:18) p log Γ(( p + n ) / k/ n/ p + k ) / (cid:19) ≤ . (3.10)Since the contribution of this term is insignificant relative to the second one, we simplyuse (3.10) as an upper bound. For the second term, for any q = 0 having the same signas p and such that k + p + q >
0, we estimate using Jensen’s inequality: ddp (cid:18) p log Γ( k + p )Γ( k ) (cid:19) = 1 pq R ∞ log( t q ) t p + k − exp( − t ) dt Γ( p + k ) − p log Γ( k + p )Γ( k ) ≤ pq log Γ( k + p + q )Γ( k + p ) − p log Γ( k + p )Γ( k ) = 1 p log Γ( k + p + q ) /q Γ( k + p ) /q Γ( k ) /p Γ( k + p ) /p ! . q = ( p + k − pk − , which indeed satisfies the aboverestrictions since p ≥ − k − , and using the latter condition on p , one verifies that: ddp (cid:18) p log Γ( k + p )Γ( k ) (cid:19) ≤ Ck ; (3.11)see also Remark 3.3 below for an alternative derivation. Plugging our estimates for L k,q obtained in Corollary 2.4, and noting that k A k op ≥ k A k HS = n , we concludethat if X is ψ α ( α ∈ [1 , b α , then: ddp log(( E | Y | p ) p ) ≤ C b α k A k op k /α p n + 1 k ! = C k /α p η + 1 k ! , (3.12)for all integers k in [max(2 , | p | + 1) , n ]. Optimizing on k in that range, we set: k = ⌈| p | /β η / (2 β ) ⌉ , β := 1 + 1 α , which is guaranteed to be in the desired range whenever | p | ∈ [4 η − / , η α/ ], as maybe easily verified using that k A k op ≥ b α ≥ − /α . Consequently, for such p , weobtain: ddp log(( E | Y | p ) p ) ≤ C | p | /β η / (2 β ) . Setting p := 4 η − / , we may assume that p ≤ η was assumed in the Introductionto be large enough (otherwise the statement of Theorem 3.1 follows easily), and sointegrating over p and adjusting constants, we obtain:exp − C (cid:18) | p − | η α (cid:19) α +1 ! ≤ ( E | Y | p ) p ( E | Y | ) ≤ exp C (cid:18) | p − | η α (cid:19) α +1 ! ∀ p ∈ [ p , η α/ ] , (3.13)and: ( E | Y | p ) p ( E | Y | − p ) − p ≥ exp − C (cid:18) | p − p | η α (cid:19) α +1 ! ∀ p ∈ [ − η α/ , − p ] . (3.14) It remains to bridge the gap between the p and − p moments. Note that since weassume that p ≤ n is larger than some constant, then p ≤ k − for e.g. k = 5. Unfortunately, in the range p ∈ [ − p , p ], our key estimate (3.12) only yields(using k = k ): ddp log(( E | Y | p ) p ) ≤ Cp η , (3.15)which in particular is not integrable at 0. We consequently treat this gap differently, byreproducing Fleury’s argument from [15]. 19ote that by Borell’s concavity result (1.18), we have: h k ,p ( u ) h k , − p ( u ) ≤ (Γ( k + p )Γ( k − p )) Γ( k ) h k , ( u ) ≤ (1 + C p ) h k , ( u ) . Taking expectation, denoting by C ov the covariance, and using the Cauchy–Schwarzinequality, we obtain:(1 + C p ) ≥ E U h k ,p ( U ) E U h k , − p ( U ) + C ov U ( h k ,p ( U ) , h k , − p ( U )) ≥ E U h k ,p ( U ) E U h k , − p ( U ) − r V ar U ( h k ,p ( U )) V ar U ( h k , − p ( U ))= E U h k ,p ( U ) E U h k , − p ( U ) − (cid:18) E U h k ,p ( U ) − ( E U h k ,p ( U )) (cid:19) (cid:18) E U h k , − p ( U ) − ( E U h k , − p ( U )) (cid:19) . Using the reverse H¨older inequality (1.20) for comparing the L / and L norms of h k ,p and h k , − p , we obtain:(1+ C p ) ≥ ( E U h k ,p ( U ) E U h k , − p ( U )) / exp − C L k ,p + L k , − p n ! − C L k ,p L k , − p n ! . By Corollary 2.4 we know that L k ,p , L k , − p ≤ C k A k op b α k /α +1 / , and we concludethat: ( E U h k ,p ( U )) p ( E U h k , − p ( U )) − p ≤ (cid:18) C η (cid:19) p ≤ C √ η . Finally, using (3.2), (3.3) and (3.10), we see that:( E | Y | p ) p ( E | Y | − p ) − p ≤ ( E U h k ,p ( U )) p ( E U h k , − p ( U )) − p ≤ C √ η . This fills the remaining gap, and together with (3.13) and (3.14), the assertion of Theorem3.1 follows.
Remark 3.2.
Examining the proof in the case α = 1 and A = Id , it is easy to verifythat if the log-Lipschitz constant L k,p of h k,p : SO ( n ) → R + satisfies:2 ≤ p ≤ k ⇒ L k,p ≤ Cp β k γ , β, γ ∈ R , then the sharp large-deviation estimate P ( | X | ≥ C √ n ) ≤ exp( −√ n ) is recovered if andonly if β + γ = 3 /
2. Of course, since p ≤ k , it is better to have larger β , and this affectsthe resulting thin-shell estimate. Our estimates yield β = 0 and γ = 3 /
2. The wastefulbound (3.15) when p is close to 0 perhaps suggests that we should expect to have β = 1and γ = 1 /
2. 20 emark 3.3.
It is possible to avoid the delicate calculation based on Stirling’s formulaleading to the bound (3.11), by replacing Borell’s concavity result (1.18) in our derivationabove by a slightly weaker concavity result due to Bobkov [8]. It states that for any log-concave density w on R + : q log R ∞ t q w ( t ) dtq q is concave on R + . A completely standard consequence of Theorem 3.1 is the following:
Theorem 4.1.
With the same assumptions and notation as in Theorem 3.1: P ( | Y | ≥ (1 + t ) √ n ) ≤ exp( − cη α min( t α , t )) ∀ t ≥ , (4.1) and: P ( | Y | ≤ (1 − t ) √ n ) ≤ C exp( − cη α max( t α , log c − t )) ∀ t ∈ [0 , . (4.2)For completeness, we provide a proof. Proof.
Set: ε η,α := min , α +2 α +1 Cη α α +1) ! , and note that there exists a constant t ∈ (0 , ∀ t ∈ ( ε η,α , t ] ∃ p ∈ (4 , cη α/ ] such that t = 2 C ( p − α +1 η α α +1) , (4.3) ∃ p ∈ [ − cη α/ ,
0) such that t = 2 C | p − | α +1 η α α +1) . (4.4)Here c, C > (cid:18) − t (cid:19) √ n ≤ ( E | Y | p ) p ≤ ( E | Y | p ) p ≤ (cid:18) t (cid:19) √ n . Since t t/ ≥ t/ t ∈ [0 , P ( | Y | ≥ (1 + t ) √ n ) ≤ P ( | Y | ≥ (1 + t/ E | Y | p ) p ) ≤ (1 + t/ − p ≤ exp( − p t/ . Expressing p as a function of t for t in the range specified in (4.3), and plugging thisabove, we obtain: P ( | Y | ≥ (1 + t ) √ n ) ≤ exp( − c η α/ t α ) ∀ t ∈ [ ε η,α , t ] .
21o extend this estimate to the entire interval [0 , t ], note that: P ( | Y | ≥ (1 + t ) √ n ) ≤ (1 + t ) − ≤ exp( − t/ ∀ t ∈ [0 , ε η,α ] , and so adjusting the constants appearing above: P ( | Y | ≥ (1 + t ) √ n ) ≤ exp( − c η α/ t α ) ∀ t ∈ [0 , t ] . Finally, a standard application of Borell’s lemma [12] (e.g. as in [34]), ensures that: P ( | Y | ≥ (1 + t ) √ n ) ≤ exp( − c η α/ t ) ∀ t ≥ t , concluding the proof of the positive deviation estimate (4.1).Similarly: P ( | Y | ≤ (1 − t ) √ n ) ≤ P ( | Y | ≤ (1 − t/ E | Y | p ) p ) ≤ (1 − t/ − p ≤ exp( p t/ . Expressing p as a function of t for t in the range specified in (4.4), and plugging thisabove, we obtain: P ( | Y | ≤ (1 − t ) √ n ) ≤ C exp( − cη α/ t α ) ∀ t ∈ [ ε η,α , t ] . Adjusting the value of C above, the estimate extends to the entire range t ∈ [0 , t ].Lastly, setting p = − c η α so that:( E | Y | p ) p ≥ √ n , we obtain for all ε ∈ (0 , / P ( | Y | ≤ ε √ n ) ≤ P ( | Y | ≤ ε ( E | Y | p ) p ) ≤ (2 ε ) − p = exp (cid:18) − c η α log (cid:18) ε (cid:19)(cid:19) . Adjusting all constants, the negative deviation estimate (4.2) follows.To conclude the proof of Theorems 1.1 and 1.2, we estimate the deviation of AX bythat of Y exactly like Klartag [23]. Indeed, according to the argument described in theproof of [23, Proposition 4.1], we have: P ( | AX | ≥ (1 + t ) √ n ) ≤ C P | AX + G n |√ ≥ r (1 + t ) + 12 √ n ! , and: P ( | AX | ≤ (1 − t ) √ n ) ≤ C P | AX + G n |√ ≤ r (1 − t ) + 12 √ n ! , for some universal constant C >
1. The deviation estimate (1.7) of Theorem 1.1 im-mediately follows from the corresponding estimates of Theorem 4.1. However, the more22efined deviation estimates (1.10) and (1.11) do not follow: (1.10) only follows up to theunnecessary constant C in front of the estimate: P ( | AX | ≥ (1 + t ) √ n ) ≤ C exp( − cη α min( t α , t )) ∀ t ≥ , (4.5)and (1.11) follows without the decay to 0 as t → P ( | AX | ≤ (1 − t ) √ n ) ≤ C exp( − cη α t α ) ∀ t ∈ [0 , . (4.6)To resolve these last issues, we proceed as follows. The unnecessary constant C > p ≥
1, by the symmetry and independence of G n , convexity of t t p and theCauchy–Schwarz inequality, we have: E | Y | p = E (cid:18) | AX + G n | (cid:19) p = 12 E (cid:18)(cid:18) | AX + G n | (cid:19) p + (cid:18) | AX − G n | (cid:19) p (cid:19) ≥ E (cid:18) | AX | + | G n | (cid:19) p ≥ E | AX | p | G n | p = E | AX | p E | G n | p ≥ E | AX | p ( E | G n | ) p/ = n p/ E | AX | p . Since E | AX | = E | Y | = k A k HS = n , we deduce:( E | AX | p ) p ( E | AX | ) ≤ ( E | Y | p ) p ( E | Y | ) ! ∀ p ≥ . (4.7)Consequently, the p -moment estimates of Theorem 3.1 hold equally true (after adjustingconstants) with Y replaced by AX , when p ≥
3. In particular, the p -moment estimates(1.14) of Theorem 1.2 for p ≥ c η α α +2) are obtained. Repeating the relevant parts inthe proof of Theorem 4.1, the desired positive deviation estimate (1.10) follows. Finally,applying [14, Lemma 6] to the deviation estimates of Theorem 1.1, the positive p -momentestimates are improved in the range 1 ≤ p − ≤ c η α α +2) , obtaining the right-hand sideof (1.13); see also below for a sketch of an alternative derivation. This takes care of thepositive moment and deviation estimates.Reducing from AX to Y the small-ball estimate (1.11), or equivalently, the negativemoment estimates of (1.14), seems more involved, and further arguments are needed.We choose to bypass these here by simply employing Paouris’ small-ball estimate (1.5),which together with (4.6) yields for some c ≤ P ( | AX | ≤ (1 − t ) √ n ) ≤ C exp( − c η α max( t α , log c − t )) ∀ t ∈ [0 , . (4.8)The negative moment estimates of (1.13) and (1.14) then follow by integrating (4.8)by parts. Since the computation is not entirely straightforward when 1 ≤ | p − | ≤ c η α α +2) , we sketch the argument, which is based on Fleury’s derivation in [14, Lemma23] of positive moment estimates from deviation estimates. However, Fleury’s techniquedoes not seem to generalize to negative moments, and so we provide an alternative proof,which is equally applicable to both positive and negative moments.Denote Z = | AX | / √ n , and note that 1 = E Z = R ∞ P ( Z ≥ t ) dt . We consequentlyhave for p > E Z − p = p Z ∞ t − ( p +1) P ( Z ≤ t ) dt = p Z P ( Z ≤ t )( t − ( p +1) − dt + p Z (1 − P ( Z ≥ t )) dt + p Z ∞ (1 − P ( Z ≥ t )) t − ( p +1) dt = p Z ∞ t − ( p +1) dt + p Z P ( Z ≤ t )( t − ( p +1) − dt + p Z ∞ P ( Z ≥ t )(1 − t − ( p +1) ) dt = 1 + p Z P ( Z ≤ − s )((1 − s ) − ( p +1) − ds + p Z ∞ P ( Z ≥ s )(1 − (1 + s ) − ( p +1) ) ds . Assuming for simplicity that p ≥
2, we use (4.8) to bound the first integral above,evaluating separately the intervals [1 − c / , , /p ] and [1 /p, − c / , /p ] and [1 /p, ∞ ).Using the obvious estimates: P ( Z ≤ − s ) ≤ P ( Z ≤ − s/ , P ( Z ≥ s ) ≤ P ( Z ≥ c min( s, s / )) ∀ s ≥ − s ) − ( p +1) − ≤ ( Cps s ∈ [0 , /p ]exp( Cps ) s ∈ [1 /p, / , − (1+ s ) − ( p +1) ≤ ( ( p + 1) s s ∈ [0 , /p ]1 s ∈ [1 /p, ∞ ) , we obtain: E Z − p ≤ pC Z c / ( ε/c ) c η α ε − ( p +1) dε (4.9)+ p C Z /p s exp( − c η α s α ) ds + pC Z − c / /p exp( − c η α s α + c ps ) ds + p C Z /p s exp( − c η α s α ) ds + pC Z ∞ /p exp( − c η α min( s α , s )) ds . When 2 ≤ p ≤ c η α α +2) , this implies using (1 + 2 px ) p ≤ x : (cid:0) E Z − p (cid:1) p ≤ C − c η α + pC Z ∞ s exp( − c η α s α ) ds + C Z ∞ /p exp( − c η α s α + c ps ) ds + C Z ∞ /p exp( − c η α min( s α , s )) ds . In this range of values for p , 1 /p ≥ c ( p/η α ) α , and hence the integrand in the terminvolving C is monotone decreasing. A standard computation then confirms that, in24his range, both integrals involving C and C are dominated by the one involving C ,yielding the negative moment estimates of (1.13); a similar argument does the job inthe positive moment range. When c η α α +2) ≤ p ≤ c η α/ , we similarly verify from (4.9)that: (cid:0) E Z − p (cid:1) p ≤ C pη α α +2) + pC Z ∞−∞ exp( − c η α | s | α + c ps ) ds ! p . Bounding the second (dominant) term using the Laplace method, we obtain the negativemoment estimates of (1.14), thereby concluding the proof of Theorem 1.2.
Appendix
In the Appendix, we prove several properties of the bodies Z + q ( K ) (for q ≥
1) which areneeded for the results of Section 2.Our main goal is to establish Proposition 2.6. For the proof, we require severallemmas. Given θ ∈ S m − , we denote H + θ := { x ∈ R m ; h x, θ i ≥ } . Lemma A.1.
Let K denote a convex body in R m , and given θ ∈ S m − , denote f θ = π θ K . Then: (cid:18) f θ (0) k f θ k ∞ (cid:19) /q (cid:18) Γ( m )Γ( q + 1)Γ( m + q + 1) (cid:19) /q h K ( θ ) ≤ h Z + q ( K ) ( θ )(2 Vol ( K ∩ H + θ )) /q ≤ h K ( θ ) . Proof.
The right inequality is straightforward from the definitions. The left inequality isderived by following the proof of [33, Lemma 4.1], which uses the fact that the 1 / ( m − K is a concave function.To control the left-most term in Lemma A.1, we have: Lemma A.2.
Let µ = f ( x ) dx denote a log-concave probability measure on R . Then forany ε > : ε ≤ Z ∞ f ( x ) dx ≤ − ε ⇒ f (0) ≥ ε k f k ∞ . This is essentially folklore (see e.g. [17, Lemma 1.1]), but we include a proof for complete-ness. We refer the interested reader e.g. to [17] for the study of functional inequalitiesin the case of non-symmetric log-concave measures.
Proof.
Let F ( x ) = R x −∞ f ( t ) dt and G ( x ) = 1 − F ( x ) = R ∞ x f ( t ) dt . By the Pr´ekopa–Leindler Theorem, both F and G are log-concave. Equivalently, this means that both f /F and − f /G are non-increasing. Consequently f ( x ) ≤ f ( y ) max( F ( x ) /F ( y ) , G ( x ) /G ( y ))for all x, y ∈ R . Using y = 0 and the assumption that F (0) , G (0) ≥ ε , the conclusionimmediately follows.This reduces our task to showing: 25 emma A.3. If w is a log-concave function on R m with barycenter at the origin, then: ∀ θ ∈ S m − (cid:18) Vol ( K m + q ( w ) ∩ H + θ ) Vol ( K m + q ( w )) (cid:19) /q ≥ c > . Proof.
Note that we may normalize and rescale so that w (0) = 1 and R R m w ( x ) dx = 1.Using polar-coordinates, we have for any convex (in fact, star-shaped) body K containingthe origin: Vol( K ∩ H + θ ) = 1 m Z S m − ∩ H + θ k ξ k − mK dξ . (A.1)Using (2.7), we see that: ∀ ξ ∈ S m − e − mqm + q k ξ k − mK m ( w ) ≤ k ξ k − mK m + q ( w ) ≤ Γ( m + q + 1) mm + q Γ( m + 1) k ξ k − mK m ( w ) . Plugging this into (A.1) and using Stirling’s formula, we verify that: ∀ θ ∈ S m − e − q ≤ Vol( K m + q ( w ) ∩ H + θ )Vol( K m ( w ) ∩ H + θ ) ≤ C q . (A.2)Using (A.1), the definition of K m ( w ) and polar-coordinates again, we see that Vol( K m ( w ) ∩ H + θ ) = R H + θ w ( x ) dx = P ( W ≥ W is the random variable on R having density π θ w . Since this density is log-concave by the Pr´ekopa–Leindler Theorem, and since thebarycenter of W is at the origin, Lemma 2.2 implies that:Vol( K m ( w ) ∩ H + θ )Vol( K m ( w )) ≥ e . (A.3)Now decomposing Vol = Vol | H + θ + Vol | H + − θ , (A.2) and (A.3) imply the assertion. Proof of Proposition 2.6.
Applying Lemma A.1 with K = K m + q ( w ) and using LemmaA.3, we obtain for all θ ∈ S m − : c (cid:18) f θ (0) k f θ k ∞ (cid:19) /q (cid:18) Γ( m )Γ( q + 1)Γ( m + q + 1) (cid:19) /q ≤ Vol( K m + q ( w )) − /q h Z + q ( K m + q ( w )) ( θ ) h K m + q ( w ) ( θ ) ≤ C .
Lemma A.2 together with Lemma A.3 imply that: ∀ θ ∈ S m − (cid:18) f θ (0) k f θ k ∞ (cid:19) /q ≥ c ′ > , and hence: c ′′ (cid:18) Γ( m )Γ( q + 1)Γ( m + q + 1) (cid:19) /q K m + q ( w ) ⊂ Vol( K m + q ( w )) − /q Z + q ( K m + q ( w )) ⊂ CK m + q ( w ) . Rearranging terms, the assertion of Proposition 2.6 follows.26inally, we prove:
Lemma A.4. If g : R m → R + is a log-concave isotropic density then Z +2 ( g ) ⊃ cB m .Proof. Given θ ∈ S n − , denote g := π θ g ; as usual, it is an isotropic log-concave proba-bility density on R . Comparing moments using the left-hand side of (2.7) with m = 1, q = 1 and q = 3, we obtain:3 Z ∞ t g ( t ) dt ≥ (cid:0)R ∞ g ( t ) dt (cid:1) e g (0) . (A.4)Applying now the reverse comparison using the right-hand side of (2.7) for both directions θ and − θ , and summing the resulting estimates, we obtain:3 = 3 Z ∞−∞ t g ( t ) dt ≤ Γ(4) g (0) (cid:18)Z ∞ g ( t ) dt (cid:19) + (cid:18)Z −∞ g ( t ) dt (cid:19) ! . (A.5)Since the barycenter of g is at the origin, we know by Lemma 2.2 that: Z −∞ g ( t ) dt ≤ ( e − Z ∞ g ( t ) dt , and so we conclude from (A.5) that: (cid:0)R ∞ g ( t ) dt (cid:1) g (0) ≥ e − ) . Together with (A.4), the assertion follows with e.g. c = (3 e (1 + ( e − )) − / . References [1] M. Anttila, K. Ball, and I. Perissinaki. The central limit problem for convex bodies.
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