\documentclass [12pt] {amsart} \headheight=6.15pt \textheight=574pt \textwidth=432pt \oddsidemargin=18.88pt \evensidemargin=18.88pt \topmargin=14.21pt \title{Covering numbers and ``low $M^{*}$-estimate'' for quasi-convex bodies} \author{A. E. Litvak} \thanks{Litvak's and Milman's research was partially supported by BSF. Research at MSRI is supported in part by NSF grant DMS-9022140.} \address{\hskip-\parindent A. E. Litvak, Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel} \email{alexandr@math.tau.ac.il} \author{V. D. Milman} \address{\hskip-\parindent V. D. Milman, Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel} \email{vitali@math.tau.ac.il} \author{A. Pajor} \address{\hskip-\parindent A. Pajor, Universite de Marne-la-Vall\'ee, Equipe de Mathematiques, 2 rue de la Butte Verte, 93166 Noisy-le-Grand Cedex, France} \email{pajor@math.univ-mlv.fr} \def\kkk{\qed} \def\ee{ \eps _0/C^p } \def\l{ \left} \def\r{ \right} \def\nek{,\ldots,} \def\eps{\varepsilon} \def\alp{\alpha} \def\ome{\Omega} \def\lam{\lambda} \def\Ker{{\rm Ker}} \def\hb{\hfill\break} \def\nor{\parallel} \def\str{\longrightarrow} \def\tt{\theta} \def\P{C(\alpha ,p) } \def\Q{C(\gamma ,q) } \def\we{we } \def\We{We } \def\bb{ J.Bastero, J.Bernu\'es, and A.Pe\~na } \newcommand{\RR}{\mbox{\rm $~\vrule height6.5pt width0.5pt depth0.3pt\!\!$R}} \newcommand{\R}{\RR ^n} \newcommand{\Rk}{\RR ^k} \newcommand{\Rm}{\RR ^m} \newcommand{\RP}{\RR ^{+}} %\def\R{ {\rm R}^n } %\def\Rm{ {\rm R}^m } %\def\Rk{ {\rm R}^k } \newcommand{\Z}{\mbox{\rm $~\vrule height6.5pt width0.5pt depth0.3pt\!\!$Z}} \newcommand{\N}{\mbox{\rm $~\vrule height6.5pt width0.5pt depth0.3pt\!\!$N}} \def\fk{ finite-dimensional version of Krivine's Theorem} \def\bm{ Brunn-Minkowski inequality } \def\pc{ $ p-$convex } \def\qs{ quasi-norm} \def\qn{ $ q-$norm} \def\no{ \nor . \nor } \def\nx{ \nor x \nor } \def\ny{ \nor y \nor } \def\nxy{ \nor x+y \nor } \def\na{ \nor a \nor } \def\nao{ \nor a \nor _0 } \def\nap{ \nor a \nor _p } \def\naoq{ \nor a \nor _0^q } \def\nua{ \nu _a(\eps ) } \def\fa{ f_a } \def\nuf{ \nu _{\fa } (\eps ) } \def\tp{ (t,\pi ) } \def\me{ \mu \{(t,\pi ) \ : \ | \fa ^q ( \tp ) } \def\bl{ balls} \def\ex { \exp (-\eps ^2 n /64 ) } \def\o{ \tt } \def\tp{ (t,\pi ) } \def\tp{ (t,\pi ) } \renewcommand{\thefootnote}{\fnsymbol{ \ }} \begin{document} %\date{} \begin{abstract} % %%%*** We give estimates on covering numbers and This article gives estimates on covering numbers and diameters of random proportional sections and projections of symmetric quasi-convex bodies in $\R$. %%%*** These results were known in convex case and played These results were known for the convex case and played an essential role in development of the theory. Because duality relations can not be applied in the quasi-convex setting, new ingredients were introduced that give new understanding %%%*** also in convex case. for the convex case as well. % %% % \end{abstract} \maketitle %00000000000000000000000000000000000000000000000000000000000000000000 %\begin{description} %00000000000000000000000000000000000000000000000000000000000000000000 % \medskip \section{Introduction and notation} %\item[{ 1. Introduction and notation. }] %\end{description} Let $ |\cdot | $ be % \hfill $ \blacksquare $ \\ a euclidean norm %\renewcommand{\thefootnote}{\fnsymbol{ \ }} %%%%%%%**********************************%%%%%%%%%%\footnote{ Supported in part by BSF.} on $\R $. Let $D$ be an ellipsoid associated with this norm. Denote $ A= \sqrt{\frac{n}{k}} \int \limits_{S^{n-1}} \sqrt{ \sum _{i=1}^{k} x_i^2 } %( \sum _{i=1}^{k} x_i^2 )^{1/2} \ d\sigma (x) $, where $\sigma $ is the normalized rotation invariant measure on the euclidean sphere $ S^{n-1} $. Then $ A=A(n,k)<1$ and $ A \str 1 $ as $ n,k \str \infty $. For any star-body $K$ in $\R$ define $ M_K= \int \limits_{S^{n-1}} \nx d\sigma (x) $, where $\nx$ is the gauge of $K$. Let $M^{*}_K$ be $M_{K^0}$, where $K^0$ is the polar of $K$. For any subsets $K_1, K_2 $ of $\R$ denote by $N(K_1,K_2)$ the smallest number $N$ such that there are %********************************************************* N points $y_1,...,y_N$ in $K_1$ such that %%$ x_i \in B \ \ (1 \leq i \leq 2^{k-1} ) $ such that $$ K_1 \subset \bigcup _{i=1}^{N} (y_i+ K_2 ) .$$ Recall that a body $K$ is called quasi-convex if there is a constant $c$ such that $K+K\subset cK$, and given a $ p \in (0,1] $ a body $K$ is called $p$-convex if for any $\lam ,\mu >0$ satisfying $\lam^p + \mu^p =1$ and any points $x,y\in K$ the point $\lam x + \mu y$ belongs to $K$. Note that for the gauge $\nor \cdot \nor \, = \, \nor \cdot \nor _K $ associated with the the quasi-convex ($p$-convex) body $K$ the following inequality holds for any $x, y \in \R $ $$ \nxy \leq C \max\{ \nx ,\ny \} \ \ \ % \l( \nxy \leq 2^{1/p}( \nx +\ny )\ \r) . $$ \l( \nxy ^p \, \leq \, %2^{1/p} \nx ^p +\ny ^p \, \r) . $$ % Of course In particular every $p$-convex body $K$ is also quasi-convex one and $K+K\subset 2^{1/p}K$. % %********************************************** % % A %\len %\footnoterule{ Supported in part by BSF.} \newpage more delicate result is that for every quasi-convex body $K$ ( $K+K\subset cK$) there exists a $q$-convex body $K_0$ such that $K\subset K_0 \subset 2c K$, where $2^{1/q}=2 c$. This is Aoki-Rolewicz theorem ([KPR], [R], see also [K], p.47). In this note by a body we always mean a centrally-symmetric compact star-body, i.e. a body $K$ satisfying $tK \subset K$ for any $t\in %(0,1) [-1, 1]$, however not all results use symmetry. Lemma~2, some analog of Lemma~4 for $p$-convex body $K$, %of A.~Pajor and Theorem~3 hold in non-symmetric case also. %*********************************************************** % DEFINITIONS %******************************************************************************** Let us remind of the so-called ``low $M^{*}$-estimate" result. % of the second named %author. % ([M], see also [PT]). \newtheorem{theorem}{Theorem} \begin{theorem} Let $\lam > 0 $ and $n$ be large enough. %($n> c/(1-\lam )^2 $). Let $K$ be a convex body in $\R$ and $\nor \cdot \nor$ be the gauge of $K$. Then there exists a subspace $E$ of $( \R , \nor \cdot \nor ) $ such that $ \dim E = [\lam n ] $ and for any $x \in E $ the following inequality holds $$ \nx \geq \frac { f(\lam ) } %{\sqrt{1-\lam }} { %c \ M^{*}_K} | x | \ \ $$ %where $c$ is an absolute constant. for some function $ f(\lam ) $, $0<\lam <1 $. \end{theorem} {\it Remark.} Inequality of this type was first proved in [M1] with very poor dependence on $\lam $ and then improved in [M2] to $ f(\lam )=C (1-\lam ) $. It was later shown ([PT]), that one can take $ f(\lam )=C \sqrt{1-\lam }$ (for different proofs see [M3] and [G]). By duality this theorem is equivalent to the following \medskip theorem. %\theoremstyle{change} %\newtheorem{claim}{Proposition } %\begin{claim} \newline \bf Theorem 1' \it Let $\lam > 0 $ and $n$ be large enough. %($n> c/(1-\lam )^2 $). For any convex body $K$ in $\R$ there exists an orthogonal projection P of %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX %the rank $[\lam n ]$ such that $$ PD \subset \frac{ %c\ M_K} %{\sqrt{1-\lam }} { f(\lam )} PK \ , $$ where $c$ is an absolute \medskip constant. %\end{claim} \rm In this note we will extend %in this note both theorems to quasi-convex bodies. Because duality arguments can not be applied to a non-convex body these two theorems become different statements. Also ``"$M^*_K$" should be substituted by an appropriate quantity not involving duality. %depending % on $K^0$. Note that by avoiding the use of convexity assumption \medskip we in fact simplified proof also for a convex case. %Since polar of body $K$ is equal to polar of $convK$, % duality argument can not be used %in a non-convex case. %So, this two theorems become non-equivalent and, to receive %such kind of results, we have to use some other technique. %\newline % %00000000000000000000000000000000000000000000000000000000000000 \section{Main results} %00000000000000000000000000000000000000000000000000000000000000 %\item[{ 2. Main results. }] The following theorem is an extension of Theorem 1'. \begin{theorem} Let $\lam > 0 $ and $n$ be large enough ($n> c/(1-\lam )^2 $). For any $p$-convex body $K$ in $\R$ there exists an orthogonal projection P of the rank $[\lam n ]$ such that $$ PD \subset \frac{A_p M_K}{(1-\lam )^{1+1/p}} PK \ , $$ %where $A_p$ depends on $p$ only. where $ A_p = const ^{\frac{\ln (2/p)}{p}}$. \end{theorem} The proof of this theorem is based on the next three lemmas. The first one was proved by W.B.Johnson and J.Lindenstrauss in [JL]. The second one was proved in [PT] for convex bodies %([PTJ]) and is the dual form of Sudakov minoration theorem. %can easily check that the %same methods work for quasi-convex bodies. \newtheorem{lemma}{Lemma} \begin{lemma} There is an absolute constant $c$ such that if $\eps > \sqrt{c/k} $ and $N< e^{\eps^2 k/c } $, then for any set of points $y_1, ... , y_N \in \R $ and any orthogonal projection P of rank $k$ $$\mu \l( \{ U\in O_n \ | \ \forall j : A(1-\eps ) \sqrt{k/n} \ |y_j | \leq | P U y_j | \leq A(1+\eps ) \sqrt{k/n} \ | y_j | \} \r) \geq $$ $$ \geq 1 - \sqrt{\pi /2} \cdot e^{-\eps ^2 k /c} $$ % Here $c$ is an absolute constant. \end{lemma} \begin{lemma} Let $K$ be a body such that $K-K \subset aK $. Then $N(D, tK) \leq 2 e^{2 n (a M_K / t)^2 }$. %where $ c_p \leq c^{1/p } $ % for some %absolute % constant $c$ \end{lemma} M.~Talagrand gave a direct simple proof of this lemma for a convex case ([T]). Below, using his idea, we prove more general lemma for $p$-convex bodies, so we do not prove Lemma 2 now. %%!!!!!!!!!! We will not prove Lemma~2, but will prove more general % % %%!!!!!!!!!! lemma for quasi-convex bodies below. % Below we present a more general lemma of A.~Pajor ([P]) for $p$-smooth %bodies which is trivially applyed for $p$-convex bodies. So we do not prove %Lemma 2 now. \begin{lemma} Let $B$ be a star-body, $K$ be a $p$-convex set, $r \in (0,1) $, $\{x_i\} \subset rB$ and $B \subset \bigcup %_{i=1}^{2^{k-1}} (x_i+K ) $. Then $ B\subset t_r K$, where $t_r= \frac{1}{(1-r^p)^{1/p}} $. \end{lemma} {\it Proof: } Obviously $t_r=\max \{ \nx _K \ | \ x\in B \} $. Since $B \subset \bigcup %_{i=1}^{2^{k-1}} (x_i+K ) $, for any point $x$ in $B$ there are points $x_0$ in $rB$ and $y$ in $K$ such that $x=x_0+y$. Then by maximality of $t_r$ and $p$-convexity of $K$ we have $t_r^p \leq r^p t_r^p + 1$. That proves the lemma. \kkk % $\Box $ \newline {\it Remark. } Somewhat similar argument was %often used by N.~Kalton in dealing with $p$-convex sets. % See, e.g. [GK]. \newline \newline % % %************************************************ % % {\it Proof of Theorem 2: } \newline Any $p$-convex body $K$ satisfies $K-K \subset aK$ with $a=2^{1/p}$. By Lemma~1 and Lemma~2, we obtain for $c_p \approx 2^{1/p} $ that if %for $c_p \approx 2^{1/p} $ $$ c_p n \l( \frac{M_K }{t} \r) ^{2} \leq \frac{\eps ^2 k}{c} $$ %($c_p$ depend on $p$ only) and $\eps > \sqrt{c/k} $, then there exist points $x_1,...,x_N$ in $D$ and an orthogonal projection $P$ of %XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX %the rank $k$ such that $$ PD \subset \bigcup (Px_i + tPK) \ \mbox{ and } \ \ |Px_i| \leq (1+\eps ) \sqrt{ \frac{k}{n} } |x_i| \ \ .$$ Let $\lam = k/n$. Denote $r=(1+\eps ) \sqrt{ \lam } $. Lemma 3 gives us $$ PD \subset t t_r PK \ \mbox{ for } \ t=\frac{\sqrt{c c_p } M_K}{\eps \sqrt{\lam }} \ \mbox{ and } \ \eps ^2 > \frac{c}{\lam n} \ , \ r<1 \ \ .$$ Choose $$ \eps = \frac{1-\sqrt{\lam }}{2 \sqrt{\lam }} \ .$$ Then for $n$ large enough %$n$ we get $$ PD \subset \frac{A_p M_K}{(1-\lam )^{1+1/p}} PK \ , $$ for $ A_p = const ^{\frac{\ln (2/p)}{p}}$ . %where $A_p$ depends on $p$ only %(in fact $ A_p = const ^{\frac{\ln (2/p)}{p}}$ ). This completes the proof. \kkk Theorem 2 can be formulated in the global \medskip form. \newline \bf Theorem 2' \it Let $K$ be a $p$-convex body in $\R$. Then there is an orthogonal operator $U$ such that \medskip $$ D \subset A^{'}_p M_K ( K+UK ) \, , \ \mbox{ where } A^{'}_p = const ^{\frac{\ln (2/p)}{p}} % \mbox{ depends on } p %\mbox{ as } A_p \mbox{ in Theorem 2. } \ .$$ \rm This theorem can be proved independently, but we show how it follows from Theorem~2. \newline {\it Proof of Theorem 2': } It follows from the proof of Theorem~2 that actually the measure of such projections is large. So we can choose two orthogonal subspaces $E_1, E_2$ of $\R$ such that $\dim E_1 = [n/2], \ \dim E_2 = [(n+1)/2] $ and $$ P_i D \subset %\frac{ A^{''}_p M_K %} %{(1-\lam )^{1+1/p}} P_i K \, , $$ where $P_i$ is the projection on the space $E_i$ ($i=1,2$). Denote $I=id_{\R }=P_1+P_2$ and $U=P_1-P_2$. So $P_1=1/2 (I+U)$ and $P_2=1/2(I-U)$. Then $U$ is an orthogonal operator and for any $x\in D$ we have $$x=P_1 x+ P_2 x \subset 1/2 \ A^{''}_p M_K (I+U) K + 1/2 \ A^{''}_p M_K (I-U) K = $$ $$ = A^{''}_p M_K \frac{K+K}{2} + A^{''}_p M_K \frac{UK-UK}{2} = A^{'}_p M_K (K+UK) \ . $$ That proves Theorem~2'. \kkk %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \rm Let us complement Lemma 2 by mentioning how %estimating covering number $N(K, tD)$ can be estimated. In convex case this estimate is given by Sudakov inequality, using quantity $M^{*}$. More precisely, if $K$ is a convex body, then $$N(K, tD) \leq 2 e^{c n ( M^{*}_K / t)^2 }\ .$$ Of course, %we mentioned above using duality for a non-convex setting leads to a weak result, and we suggest below a substitution for quantity $M^{*}$. %Let us discuss Theorem 1. Using technics of [MP1] %we can obtain this theorem %(with factor $(1- \lam )^{-1}$ istead of $(1- \lam )^{-1/2}$) % from Lemma 1 and %well known estimate of %covering number $N(K, tD) \leq e^{cn(M^{*}/t)^2 }$. %In quasi-normed case this estimate not so good, %because we can cover %$convK$ by such number of translations of $t D$. % Define the following %characteristic of $K$, that allows us estimate %covering number body $N(K, tD)$ without %using of $M^{*}_K$. %Let $$ \tilde M_K = \frac{1}{|K|} %\int \limits_{K} |x| dx \ ,$$ %where $|K|$ is volume of $K$. For two quasi-convex %$K, B $ bodies $K, B $ define the following number $$ M(K, B) = \frac{1}{|K|} \int \limits_{K} \nx _B dx \ ,$$ where $|K|$ is volume of $K$, and $\nx _B$ is the gauge of $B$. Such numbers are considered in [MP1], [MP2] and [BMMP]. % We use the following lemma of A.~Pajor ([P]). % This number does not depend on polar of $K$ and, %it seems to me, less then $M^{*}_K$. For example, % if $K=B(l^n_1)$ we %have $ \tilde M_K \leq const \ n^{-1/2}$ and %$ M^{*}_K \geq %const \ n^{-1/2} (\log n )^{1/2}$. Unfortunately, % there are examples, when $ \tilde M_K $ has the %same order as $ M^{*}_K$ % ($ K=B(l^n_{\infty })$). % % % % % %\begin{lemma} % Let $K$ be a $p$-convex body and $B$ be a %star-body such that $B + B \subset aB$. % Let $\o >0$ and $c_{\o } = 2^{1/p-1/\o } $ %if $\o >p$ and $c_{\o } =1$ %otherwise. %Then % $$N(K , tB) \leq 2 c_{\o }^n %e^{ (c n /\o ) (a M(K, B) / t)^{\o} }\ ,\ %\mbox{ where } c \mbox{ is the %absolute % constant. }$$ % %\end{lemma} %% % % \begin{lemma} Let $K$ and $B$ be % $p$-convex bodies. %and $B$ be a star-bodies. %such that Assume $B + B \subset aB$. For $\o >0$ denote by $c_{\o }= c_{\o } (K) $ %denote the best possible constant such that $$ \nor x+y \nor _K^{\o } + \nor x-y \nor _K^{\o } \leq 2 \cdot c_{\o }^{\o } \l( \nx _K^{\o } + \ny _K^{\o } \r) \ \mbox{ for any } \ x, y \in \R \ . $$ % Let $\o >0$ and $c_{\o } = 2^{1/p-1/\o } $ %if $\o >p$ and $c_{\o } =1$ %otherwise. Then $$N(K , tB) \leq 2 c_{\o }^n e^{ (c n /\o ) (a M(K, B) / t)^{\o} }\ ,\ \mbox{ where } c \mbox{ is an absolute constant. }$$ \end{lemma} Note, that in a case $\o =1$ which correspond to the general convex case this lemma was announced in [MP2]. % For completeness of proof we prove this lemma below. \newline % % (Note, that in case $\o =1$ which correspond to the general %> convex case this lemma was announced in [MP2].) % % %NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOTTTTTTTTTTT % % {\it Proof: } We follow the idea of M.Talagrand of estimating covering numbers in case $K=D$ ([T], see also [BLM] Proposition 4.2). Denote the gauge of $K$ by $\nor \cdot \nor $ and the gauge of $B$ by $ | \cdot | $. Define the measure $\mu $ by following $$d\mu =\frac{1}{A} e^{-\nx ^{\o }} dx \ , \mbox{ where } %\nx = \nx _B % \mbox{ and } \ A % % \ A \ \mbox{ chosen such that } \ \int \limits_{\R } d\mu =1 \ . $$ Let $L= %\frac{1}{A} \int \limits_{\R } |x| d\mu $. Then $\mu \{|x| \leq 2L \} \geq 1/2 $. Let $x_1,x_2,...$ be a maximal set of points in $K$ such that $ |x_i-x_j| \geq t$. So the sets $x_i+\frac{t}{a} B % \bigcap x_j+t/a int B = %\emptyset $ have mutually disjoint interiors. Let $y_i =\frac{ ab}{t} x_i $ for some $b$. Then, by symmetry of $B$ and convexity of the function $e^t$, we have $$ \mu \{ y_i+bB \} =\frac{1}{A} \int \limits_{bB } e^{-\nor x+y_i \nor ^{\o }} dx = \frac{1}{2 A} \int \limits_{bB } e^{-\nor x+y_i \nor ^{\o } } + e^{-\nor x-y_i \nor ^{\o }} dx \geq $$ $$ \geq \frac{1}{A} \int \limits_{bB } e^{-\frac{1}{2}(\nor x+y_i \nor ^{\o } +\nor x-y_i \nor ^{\o } )} dx % \frac{1}{A} = %%%%%%%%%%= %%%%%%%%%%\int \limits_{bD } e^{-(\nor x \nor %%%%%%%%%% ^{2 } + \nor y_i \nor %%%%%%%%%% ^{2 })} dx %%%%%%%%%%\ , $$ % % % % % % % % % % % % %$$ %\geq \frac{1}{A} \int \limits_{bB } e^{-((\nor x \nor % ^{p } + \nor y_i \nor % ^{p })^{\o /p}} dx \geq $$ %$$ \geq \frac{1}{A} \int \limits_{bB } e^{- c_{\o }^{\o } (\nor x \nor ^{\o } + \nor y_i \nor ^{\o })} dx = $$ $$ = \frac{1}{A} e^{-\nor c_{\o }y_i \nor ^{\o }} \int \limits_{bB } e^{-\nor c_{\o } x \nor ^{\o }} dx \geq e^{-(c_{\o } ba/t)^{\o } } c_{\o }^{-n} \mu \{ c_{\o }bB \} $$ Choose $b=2L/ c_{\o } $. Then $ \mu \{ c_{\o }bB \} \geq 1/2 $ and, hence, $$ N(K,tB) \leq 2 c_{\o }^n e^{(2 a L/t)^{\o } } \ . $$ %*********************************************************** Now compute $L$. First, the normalization constant $A$ is equal $$A = \int \limits_{\R } e^{-\nx ^{\o }} dx = \int \limits_{\R } \int \limits_{\nx }^{\infty } (-e^{-t^{\o }})' dt dx = \int \limits_{0 }^{\infty } \o t^{\o -1 } e^{-t^{\o }} \int \limits_{\nx \leq t } dx dt = $$ $$ = \int \limits_{\nx \leq 1 } dx \int \limits_{0 }^{\infty } \o t^{\o +n-1} e^{-t^{\o }}dt = |K| \cdot \Gamma \l(1+\frac{n}{\o } \r) \ ,\ %\mbox { where } \Gamma %\mbox{ is the gamma-function.} $$ where $ \Gamma $ is the gamma-function. The remaining integral is $$ \int \limits_{\R } |x| e^{-\nx ^{\o }} dx = \int \limits_{\R } |x| \int \limits_{\nx }^{\infty } (-e^{-t^{\o }})' dt dx = \int \limits_{0 }^{\infty } \o t^{\o -1 } e^{-t^{\o }} \int \limits_{\nx \leq t } |x| dx dt = $$ $$ = \int \limits_{\nx \leq 1 } |x| dx \int \limits_{0 }^{\infty } \o t^{\o +n} e^{-t^{\o }}dt = |K| \cdot M(K,B) \cdot \Gamma \l( 1+\frac{n+1}{\o } \r) \ %,\ \mbox { where } \Gamma %\mbox{ is the gamma-function} \ . $$ Using Stirling's formula we get $$L\approx \l( \frac{n}{\o } \r) ^{1/\o } M(K, B) \ .$$ That proves the lemma. \kkk % This lemma is an extension of Lemma 2. %It follows from the proof, that %in case $K=D$, %$$N(D , tB) \leq 2 e^{ (c n ) %( a M(D, B) / t)^{2} } \ ,\ \mbox{ because of }$$ %$$ \int \limits_{bD } e^{-\nor x+y_i \nor _D % ^2} dx = \frac{1}{2} % \int \limits_{bD } e^{-\nor x+y_i \nor _D^2} + e^{-\nor x-y_i \nor _D^2} dx %\geq $$ %$$ % \geq % \int \limits_{bD } %e^{-\frac{1}{2}(\nor x+y_i \nor _D^2 +\nor x+y_i \nor _D^2 )} dx %% \frac{1}{A} = %= %\int \limits_{bD } e^{-(\nor x \nor % ^{2 } + \nor y_i \nor % ^{2 })} dx %\ , $$ % and, by direct computation, % %%of course, % $M(D,B)= \frac{n}{n+1}M_B$. This lemma is an extension of Lemma 2. Indeed, since Euclidean space is a 2-smooth %a space, %of the type 2, then in case $K=D$ being an ellipsoid, we have $c_2(D) =1$. By direct computation, $M(D,B)= \frac{n}{n+1}M_B$. Thus, $$N(D , tB) \leq 2 e^{ (c n ) ( M_B %(D, B) / t)^{2} } \, .$$ Define the following characteristic of $K$, %that allows us estimate %covering number body $N(K, tD)$ without %using of $M^{*}_K$. $$ \tilde M_K = \frac{1}{|K|} \int \limits_{K} |x| dx \, .$$ %where $|K|$ is volume of $K$. By definition, if $K$ is a $p$-convex body, then for any $x, y \in \R $ holds $$ \nor x+y \nor _K^{p } + \nor x-y \nor _K^{p } \leq 2 \cdot \l( \nx _K^{p } + \ny _K^{p } \r) \, . %\mbox{ for any } \ x, y \in \R \ . $$ So, the last lemma shows that for $p$-convex body $K$ $$N(K , tD) \leq 2 e^{ (c n /p ) (2 %^{1/p} \tilde M_K / t)^{p} } \, .$$ %%% The last lemma %%% shows that for $p$-convex %%%body $K$ %%%%%%%%%%%%%%%%%%%%%%%%%allows us estimate covering number %%%$$N(K , tB) \leq %%%2 e^{ (c n /p ) (2^{1/p} \tilde M_K / t)^{p} } \ .$$ %%%%%%%%%%%%%%%%%%%%%%%%%% for $p$-convex %%%%%%%%%%%%%%%%%%%%%%%%%%%body $K$. The Theorem 3 follows from this estimate by arguments similar of that in [MP]. % As we mention above this lemma gives us Theorem 4. \begin{theorem} Let $\lam > 0 $ and $n$ be large enough. %($n> c/(1-\lam )^2 $). Let $K$ be a $p$-convex body in $\R$ and $\nor \cdot \nor$ be the gauge of $K$. Then there exists subspace $E$ of $( \R , \nor \cdot \nor ) $ such that $ \dim E = [\lam n ] $ and for any $x \in E $ the following inequality holds $$ \nx \geq \frac{(1-\lam )^{1/2+1/p}}{a_p \tilde M_K} | x | \, ,$$ where $a_p$ depends on $p$ only (more precisely $ a_p = const ^{\frac{\ln (2/p)}{p}}$). \end{theorem} {\it Proof: } By Lemma 4 there are points $x_1, ... , x_N$ in $K$, such that $N< e^{c_p n (\tilde M_K / t )^p}$ and for any $x \in K$ there exists some $x_i$ such that $|x-x_i| \sqrt{\frac{c}{\delta n}} $$ we have $$b|x_i| =(1-\eps )A \sqrt{\delta } |x_i| \leq |Px_i| \leq (1+\eps )A \sqrt{\delta } |x_i| $$ for every $x_i$. Let $E=\Ker P$. Then $\dim E=\lam n$, where $\lam = 1-\delta$. Take $x$ in $K \bigcap E $. There is $x_i$ such that $|x-x_i|0 \ | \ N(D, \eps K ) \leq 2^{k-1} \} $ . %Point out, that Theorem~2 can be obtained from this results. It is worth to point out that Theorem~2 can be obtained from this results. We thank E.~Gluskin for his remarks on the first draft of this note. \section*{References} {\footnotesize \begin{description} %\item[{[BL]}] Bergh, J.; L\"ofstr\"om, J. %{\em Interpolation spaces. An introduction.} %Springer-Verlag, 223, Berlin Heildeberg, 1976. \smallbreak \item[{[BLM]}] Bourgain, J. ; Lindenstrauss, J. ; Milman, V. {\em Approximation of zonoids by zonotopes.} Acta Math. 162 (1989), no. 1-2, 73--141. \smallbreak \item[{[BMMP]}] Bourgain, J. ; Meyer, M. ; Milman, V. ; Pajor, A. {\em On a geometric inequality.} Geometric aspects of functional analysis (1986/87), 271--282, Lecture Notes in Math., 1317, Springer, Berlin-New York, 1988. %(Reviewer: Yehoram Gordon) 46B05 (46B20) \smallbreak \item[{[JL]}] Johnson, W. B.; Lindenstrauss, J. {\em Extensions of Lipschitz mappings into a Hilbert space.} Conference in modern analysis and probability (New Haven, Conn., 1982), 189--206, %Contemp. Math., 26, Amer. Math. 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