%\input amstex \magnification=1200\overfullrule=0pt\tolerance=2000 \documentstyle{amsppt} \hcorrection{.25in} \define\({\left(} \define\){\right)} \define\[{\left\[} \define\]{\right]} \redefine\epsilon{\varepsilon} \define\R{\Bbb R^n} \define\s{\Bbb S^{n-1}} \define\e{\varepsilon} \define\la{\lambda} \define\C{C_1 (\delta)} \redefine\phi{\varphi} \define\eb{\Cal B} \topmatter \title A note on the $M^*$-limiting convolution body \endtitle \author antonis tsolomitis\endauthor %\affil MSRI and The Ohio State University \endaffil \address Department of Mathematics, 231 W.18th Avenue, Columbus, OH 43210 \endaddress \curraddr Mathematical Sciences Research Institute, 1000 Centennial Road, Berkeley, CA 94720 \endcurraddr \email atsol\@math.mps.ohio-state.edu\endemail \abstract We introduce the mixed convolution bodies of two convex symmetric bodies. We prove that if the boundary of a body $K$ is smooth enough then as $\delta$ tends to $1$ the $\delta$--$M^*$--convolution body of $K$ with itself tends to a multiple of the Euclidean ball after proper normalization. On the other hand we show that the $\delta$--$M^*$--convolution body of the $n$--dimensional cube is homothetic to the unit ball of~$\ell_1^n$. \endabstract \endtopmatter \document \head 1. Introduction \endhead \footnotetext""{Research at MSRI partially supported by NSF grant DMS-9022140.}% Throughout this note $K$ and $L$ denote convex symmetric bodies in $\R$. Our notation will be the standard notation that can be found, for example, in \cite{2} and \cite{4}. For $1\leq m\leq n$, $V_m (K)$ denotes the $m$--th mixed volume of $K$ (i.e. mixing $m$ copies of $K$ with $n-m$ copies of the Euclidean ball $\eb_n$ of radius one in $\R$). Thus if $m=n$ then $V_n (K)=\text{vol}_n (K)$ and if $m=1$ then $V_1 (K)=w(K)$ the mean width of $K$. For $0<\delta<1$ we define the $m$--th mixed $\delta$--convolution body of the convex symmetric bodies $K$ and $L$ in $\R$: \definition{Definition \rom{1.2} } The $m$--th mixed $\delta$--convolution body of $K$ and $L$ is defined to be the set, $$C_m (\delta;K,L)= \{x\in\R\ :\ V_m \( K\cap (x+L)\)\geq \delta V_m (K\cap L )\} .$$ \enddefinition It is a consequence of Brunn--Minkowski inequality for mixed volumes that these bodies are convex. If we write $h(u)$ for the support function of $K$ in the direction $u\in\s$ then we have, $$w(K)=2M_K^* =2 \int_{\s} h(u) d\nu (u),\tag1.1 $$ where $\nu$ is the Lebesgue measure of $\R$ restricted on $\s$ and normalized so that $\nu (\s )=1$. In this note we study the limiting behavior of $C_1 (\delta;K,K)$ (which we will abbreviate with $C_1 (\delta)$) as $\delta$ tends to $1$ and $K$ has a $C_+^2$ boundary. For simplicity we will call $C_1 (\delta)$ the \lq\lq $\delta$--$M^*$--convolution body of $K$". We are looking for suitable $\alpha\in\Bbb R$ so that the limit $$\lim_{\delta\rightarrow 1^- } \frac{C_1 (\delta)}{(1-\delta)^\alpha} $$ exists (convergence in the Hausdorff distance). In this case we call the limiting body \lq\lq the limiting $M^*$--convolution body of $K$". We prove that for a convex symmetric body $K$ in $\R$ with $C_+^2$ boundary the limiting $M^*$--convolution body of $K$ is homothetic to the Euclidean ball. We also get a sharp estimate (sharp with respect to the dimension $n$) of the rate of the convergence of the $\delta$--$M^*$--convolution body of $K$ to its limit. By $C_+^2$ we mean that the boundary of $K$ is $C^2$ and that the principal curvatures of $bd(K)$ at every point are all positive. %In addition to this result we compute the limiting $M^*$--convolution body %for polytopes (see Theorem $3.1$). We show that if $P$ is a convex symmetric %polytope in $\R$ then there exists a $0<\delta_0 =\delta_0 (P)<1$so that %the $\delta$--$M^*$--convolution body $C(\delta ;P)$ %is independent of $\delta$ as long as $\delta_0 \leq \delta\leq 1$. We also show that some smoothness condition on the boundary of $K$ is necessary for this result to be true, by proving that the limiting $M^*$--convolution body of the $n$--dimensional cube is homothetic to the unit ball of $\ell_1^n$. We want to thank Professor V.D.Milman for his encouragement and his guidance to this research and for suggesting the study of mixed convolution bodies. \head 2. The case \lq\lq $bd(K)$ is a $C_+^2$ manifold"\endhead In this section we prove the following: \proclaim{Theorem \rom{2.1}} Let $K$ be a convex symmetric body in $\R$ so that $bd(K)$ is a $C_+^2$ manifold. Then for all $x\in\s$ we have, $$\bigg\vert \Vert x\Vert_{\frac{C_1 (\delta)}{1-\delta}} -\frac{c_n}{M_K^*} \bigg\vert\leq C\frac{c_n}{M_K^*} \( M_K^* n (1-\delta) \)^2 ,\tag$2.1.1$ $$ where $c_n =\int_{\s} |\langle x,u\rangle | d\nu (u) \sim 1/\sqrt{n}$ and $C$ is a constant independent of the dimension $n$. In particular, $$\lim_{\delta\rightarrow 1^-} \frac{C_1 (\delta)}{1-\delta} = \frac{M_K^*}{c_n} \eb_{n} .$$ Moreover the estimate $(2.1.1)$ is sharp with respect to the dimension $n$ \endproclaim By \lq\lq sharp" with respect to the dimension $n$ we mean that there are examples (for instance the $n$--dimensional Euclidean ball) for which the inequality $(2.1.1)$ holds true if \lq\lq $\leq$" is substituted with \lq\lq $\geq$" and the constant $C$ is adjusted by a (universal) constant factor. Before we proceed with the proof we will need to collect some standard notation which can be found in \cite{4}. We write $p:\ bd(K)\rightarrow \s$ for the Gauss map $p(x)=N(x)$ where $N(x)$ denotes the unit normal vector of $bd(K)$ at $x$. $W_x$ denotes the Weingarten map, that is, the differential of $p$ at the point $x\in bd(K)$. $W_u^{-1}$ is the reverse Weingarten map at $u\in\s$ and the eigenvalues of $W_x$ and $W_u^{-1}$ are respectively the principal curvatures and principal radii of curvature of the manifold $bd(K)$ at $x\in bd(K)$ and $u\in\s$. We write $\Vert W\Vert$ and $\Vert W^{-1}\Vert$ for the quantities: $\sup_{x\in bd(K)} \Vert W_x\Vert$ and $\sup_{u\in\s} \Vert W_u^{-1} \Vert$ respectively. These quantities are finite since the manifold $bd(K)$ is assumed to be $C_+^2$. For $\la\in\Bbb R$ and $x\in\s$ we write $K_\la$ for the set $K\cap (\la x +K )$. $p_\la^{-1} :\ \s\rightarrow bd(K_\la )$ is the reverse Gauss map, that is, the affine hyperplane $p_\la^{-1} (u)+[u]^\perp$ is tangent to $K_\la$ at $p_\la^{-1} (u)$. The normal cone of $K_\la$ at $x$ is denoted by $N(K_\la ,x)$ and similarly for $K$. The normal cone is a convex set (see \cite{4}). Finally $h_\la$ will denote the support function of $K_\la$. \demo{Proof} Without loss of generality we may assume that both the $bd(K)$ and $\s$ are equipped with an atlas whose charts are functions which are Lipschitz, their inverses are Lipschitz and they all have the same Lipschitz constant $c>0$. Let $x\in\s$ and $\la=\frac1{\Vert x\Vert_{C_1 (\delta)}}$; hence $\la x\in bd\( C_1 (\delta)\)$ and $$M_{K_\la }^* =\delta M_K^* .\tag$2.1.2$ $$ We estimate now $M_{K_\la}^*$. Let $u\in\s$. We need to compare $h_\la (u)$ and $h(u)$. Set $Y_\la =bd(K)\cap bd(\la x+K)$. {\it Case 1}. $p_\la^{-1} (u)\notin Y_\la$. In this case it is easy to see that $$ h_\la (u)= h(u)-|\langle \la x,u\rangle|.$$ {\it Case 2}. $p_\la^{-1} (u)\in Y_\la$. Let $y_\la =p_\la^{-1} (u)$ and $y'_\la =y_\la -\la x\in bd(K)$. The set $N(K_\la , y_\la )\cap\s$ defines a curve $\gamma$ which we assume to be parametrized on $[0,1]$ with $\gamma (0)=N(K, y_\la)$ and $\gamma (1)= N(K, y'_\la )$. We use the inverse of the Gauss map $p$ to map the curve $\gamma$ to a curve $\tilde{\gamma}$ on $bd(K)$ by setting $\tilde{\gamma}=p^{-1} \gamma$. The end points of $\tilde{\gamma}$ are $y_\la$ (label it with $A$) and $y'_\la$ (label it with $B$). Since $u\in\gamma$ we conclude that the point $p^{-1} (u)$ belongs to the curve $\tilde{\gamma}$ (label this point by $\Gamma$). Thus we get: $$0\leq h(u)-h_\la (u)=|\langle \vec{A\Gamma}, u\rangle |.$$ It is not difficult to see that the cosine of the angle of the vectors $\vec{A\Gamma}$ and $u$ is less than the largest principal curvature of $bd(K)$ at $\Gamma$ times $|\vec{A\Gamma} |$, the length of the vector $\vec{A\Gamma}$. Consequently we can write, $$0\leq h(u)-h_\la (u)\leq \Vert W\Vert |\vec{A\Gamma}|^2 .$$ In addition we have, $$\align |\vec{A\Gamma}|&\leq \text{length}\( \tilde{\gamma}|_A^\Gamma \)\leq \text{length} \(\tilde{\gamma}|_A^B \)\\ &=\int_0^1 |d_t \tilde{\gamma} |dt=\int_0^1 |d_t p^{-1} \gamma |dt\\ &\leq\Vert W^{-1} \Vert \text{length} (\gamma)\leq\frac2{\pi} \Vert W^{-1} \Vert |p(y_\la )-p(y'_\la )|, \endalign $$ where $|\centerdot |$ is the standard Euclidean norm. Without loss of generality we can assume that the points $y_\la$ and $y'_\la$ belong to the same chart at $y_\la$. Let $\phi$ be the chart mapping $\Bbb R^{n-1}$ to a neighborhood of $y_\la$ on $bd(K)$ and $\psi$ the chart mapping $\Bbb R^{n-1}$ on a neighborhood of $N(K, y_\lambda )$ in $\s$. We assume, as we may, that the graph of $\gamma$ is contained in the range of the chart $\psi$. It is now clear from the above series of inequalities that $$|\vec{A\Gamma}|\leq c_0 \Vert W^{-1} \Vert |\psi^{-1} p \phi (t)- \psi^{-1} p \phi (s)|,$$ where $t$ and $s$ are points in $\Bbb R^{n-1}$ such that $\phi (t)=y_\la$ and $\phi (s)=y'_\la$ and $c_0 >0$ is a universal constant. Now the mean value theorem for curves gives, $$\align |\vec{A\Gamma} |&\leq C\Vert W^{-1} \Vert \Vert W\Vert |t-s|\\ &\leq C\Vert W^{-1} \Vert \Vert W\Vert |y_\la -y'_\la |\\ &=C\Vert W^{-1} \Vert \Vert W\Vert \la , \endalign $$ where $C$ may denote a different constant every time it appears. Thus we have, $$0\leq h(u)-h_\la (u)\leq C\Vert W\Vert\(\Vert W^{-1} \Vert \Vert W\Vert\)^2 \la^2 .$$ Consequently, $$ \int_{\s\setminus p_\la (Y_\la )} \( h(u)-|\langle \la x, u\rangle|\) d\nu (u) + \int_{p_\la (Y_\la )} \( h(u)-C \la^2 \) d\nu (u)$$ $$\leq M_{K_\la}^* =\delta M_K^* \leq$$ $$\int_{\s\setminus p_\la (Y_\la )} \( h(u)-|\langle \la x, u\rangle|\) d\nu (u) + \int_{p_\la (Y_\la )} h(u)d\nu (u) ,$$ where $C$ now depends on $\Vert W\Vert$ and $\Vert W^{-1} \Vert$. Rearranging and using $c_n$ for the quantity $\int_{\s} |\langle x,u\rangle | d\nu (u)$ and the fact $\la=1/\Vert x\Vert_{C_1 (\delta)}$ we get: $$\bigg\vert \Vert x\Vert_{\frac{C_1 (\delta)}{1-\delta}} -\frac{c_n}{M_K^*} \bigg\vert\leq \frac{c_n}{M_K^*} \(\frac{\int_{p_\la (Y_\la )} |\langle x,u\rangle|d\nu (u)}{c_n} + C\la\frac{\mu\(p_\la (Y_\la )\)}{c_n} \).$$ We observe now that for $u\in p_\la (Y_\la ), |\langle x,u\rangle|\leq \text{length} (\gamma)/2 \leq \Vert W\Vert \la$. Using this in the last inequality and the fact that $p_\la (Y_\la )$ is a \lq\lq band" around an equator of $\s$ of width at most $\text{length} (\gamma )/2$ we get, $$\align \bigg\vert \Vert x\Vert_{\frac{C_1 (\delta)}{1-\delta}} -\frac{c_n}{M_K^*} \bigg\vert&\leq \frac{c_n}{M_K^*} C n\la^2 \\ &\leq \frac{c_n}{M_K^*} C n\frac{(1-\delta)^2 }{\Vert x\Vert_{ \frac{C_1 (\delta)}{1-\delta}}^2 }. \endalign $$ Our final task is to get rid of the norm that appears on the right side of the latter inequality. Set $$T=\frac{\Vert x\Vert_{C_1 (\delta) /1-\delta }}{c_n /M_K^* } .$$ We have shown that $$T^2 \vert T-1 \vert \leq C \frac{M_K^*}{c_n} n (1-\delta )^2 .$$ If $T\geq 1$ then we can just drop the factor $T^2$ and we are done. If $T<1$ we write $T^2 |T-1|$ as $\(1-(1-T)\)^2 (1-T)$ and we consider the function $$f(x)=(1-x)^2 x\ :\ (-\infty ,\frac13 )\rightarrow\Bbb R.$$ This function is strictly increasing thus invertible on its range, that is, $f^{-1}$ is well defined and increasing in $(-\infty ,\frac4{27} )$. Consequently if $$C\frac{M_K^*}{c_n} n(1-\delta)^2 \leq \frac4{27} , \tag$2.1.3$ $$ we conclude that, $$\align 0\leq 1-T &\leq f^{-1} \( C\frac{M_K^*}{c_n} n(1-\delta)^2 \)\\ &\leq C\frac{M_K^*}{c_n} n(1-\delta)^2 . \endalign $$ The last inequality is true since the derivative of $f^{-1}$ at zero is $1$. Observe also that the convergence is \lq\lq essentially realized" after $(2.1.3)$ is satisfied.\qed \enddemo We now proceed to show that some smoothness conditions on the boundary of $K$ are necessary, by proving that the limiting $M^*$--convolution body of the $n$--dimensional cube is homothetic to the unit ball of $\ell_1^n$. In fact we show that the $\delta$--$M^*$--convolution body of the cube is already homothetic to the unit ball of $\ell_1^n$. \proclaim{Example \rom{2.3} } Let $P=[-1,1]^n$. Then for $0<\delta<1$ we have, $$C_1 (P)=\frac{C_1 (\delta;P,P)}{1-\delta} =n^{3/2} \text{vol}_{n-1} (\s) B_{\ell_1^n} .$$ \endproclaim \demo{Proof} Let $x=\sum_{j=1}^n x_j e_j$ where $x_j \geq 0$ for all $j=1,2,\dots ,n$ and $e_j$ is the standard basis of $\R$. Let $\la>0$ be such that $\la x\in bd\( C_1 (\delta)\)$. Then, $$P\cap (\la x+P)=\{ y\in\R :\ y=\sum_{j=1}^n y_i e_i , -1+\la x_j \leq y_j \leq 1 , j=1,2,\dots ,n \}.$$ The vertices of $P_\la =P\cap (\la x+P)$ are the points $\sum_{j=1}^n \alpha_j e_j$ where $\alpha_j$ is either $1$ or $-1+\la x$ for all $j$. Without loss of generality we can assume that $-1+\la x_j <0$ for all the indices $j$. Put $\hbox{sign}\,\alpha_j =\alpha_j /|\alpha_j |$ when $\alpha_j \neq 0$ and $\hbox{sign}\,0=0$. Fix a sequence of $\alpha_j$'s so that the point $v=\sum_{j=1}^n \alpha_j e_j$ is a vertex of $P_\la$. Clearly, $$N\( P_\la , v\)=N\(P, \sum_{j=1}^n (\hbox{sign}\,\alpha_j)e_j \).$$ If $u\in\s\cap N(P_\la ,v)$ then, $$h_\la (u)=h (u)-\bigg\vert \langle \sum\limits_{j=1}^n (\alpha_j - \hbox{sign}\,\alpha_j )e_j ,u\rangle \bigg\vert .$$ If $\hbox{sign}\,\alpha_j =1$ then $\alpha_j -\hbox{sign}\,\alpha_j =0$ otherwise $\alpha_j -\hbox{sign}\,\alpha_j =\la x .$ Let $\Cal A \subseteq \{1,2,\dots ,n \}$. Consider the \lq\lq$\Cal A$--orthant": $$\Cal O_{\Cal A} =\{y\in\R :\ \langle y, e_j \rangle <0 , \ \text{if}\ j\in\Cal A\ \text{and}\ \langle y,e_j \rangle \geq 0\ \text{if} \ j\notin\Cal A \} .$$ Then $\Cal O_{\Cal A} =N\( P, \sum_{j=1}^n (\hbox{sign}\,\alpha_j )e_j \)$ if and only if $\hbox{sign}\,\alpha_j =1$ exactly for every $j\notin\Cal A$. Hence, $$h_\la (u)=h(u)-\bigg\vert \langle\sum\limits_{j\in\Cal A} \la x_j e_j ,u \rangle \bigg\vert,$$ for all $u\in\Cal O_{\Cal A} \cap \s$. Hence using the facts $M_{P_\la}^* =\delta M_P^*$ and $\la=1/\Vert x\Vert_{ C_1 (\delta ) }$ we get, $$\Vert x\Vert_{\frac{C_1 (\delta)}{1-\delta}} =-\frac1{M_P^*} \sum\limits_{\Cal A\subseteq \{ 1,2,\dots , n\} } \sum\limits_{j\in\Cal A} x_j \int_{\Cal O_{\Cal A} \cap \s } \langle e_j ,u\rangle d\nu (u) ,$$ which gives the result since $$\int_{\Cal O_{\Cal A} \cap \s } \langle e_j ,u\rangle d\nu (u)= \frac1{2^{n-1}} \int_{\s} |\langle e_1 ,u\rangle| d\nu (u).$$\qed \enddemo %\Refs \head{References}\endhead \widestnumber\no{1} \ref \no 1 \by Kiener, K\. \paper Extremalit\"at von Ellipsoiden und die Faltungsungleichung von Sobolev \jour Arch\. Math. \vol 46 \yr 1986 \pages 162--168\endref \ref \no 2 \by Milman, V\. and Schechtmann, G\. \book Asymptotic theory of finite dimensional normed spaces \publ Springer Lecture Notes \vol 1200 \yr 1986 \endref \ref \no 3 \by Schmuckenschl\"ager, M\. \paper The distribution function of the convolution square of a convex symmetric body in $\Bbb R ^n$ \jour Israel Journal of Mathematics \vol 78 \yr 1992 \pages 309--334 \endref \ref \no 4 \by Schneider, R\. \book Convex bodies: The Brunn--Minkowski theory \publ Encyclopedia of Mathematics and its Applications, Cambridge University Press \vol 44 \yr 1993 \endref \ref \no 5 \by Tsolomitis, A\. \paper Convolution bodies and their limiting behavior \jour Duke J. Math. (submitted) \endref %\endRefs \enddocument