Related papers: A remark on the slicing problem

For any origin-symmetric convex body $K$ in $\mathbb{R}^n$ in isotropic position, we obtain the bound: \[ M^*(K) \leq C \sqrt{n} \log(n)^2 L_K ~, \] where $M^*(K)$ denotes (half) the mean-width of $K$, $L_K$ is the isotropic constant of…

We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that $$S(K)\leq\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in…

We provide a reformulation of the hyperplane conjecture (the slicing problem) in terms of the floating body and give upper and lower bounds on the logarithmic Hausdorff distance between an arbitrary convex body $K\subset \mathbb{R}^{d}$\…

We study the lower bound for Koldobsky's slicing inequality. We show that there exists a measure $\mu$ and a symmetric convex body $K \subseteq \mathbb{R}^n$, such that for all $\xi\in S^{n-1}$ and all $t\in \mathbb{R},$…

In this paper, we obtain the best possible value of the absolute constant $C$ such that for every isotropic convex body $K \subseteq \mathbb{R}^n$ the following inequality (which was proved by Klartag and reduces the hyperplane conjecture…

Let $C$ and $K$ be centrally symmetric convex bodies in ${\mathbb R}^n$. We show that if $C$ is isotropic then \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s \leq c_1L_C(\log…

A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its…

We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${\mathbb…

For $n\in \mathbb{N}$ let $S_n$ be the smallest number $S>0$ satisfying the inequality $$ \int_K f \le S \cdot |K|^{\frac 1n} \cdot \max_{\xi\in S^{n-1}} \int_{K\cap \xi^\bot} f $$ for all centrally-symmetric convex bodies $K$ in…

We provide general estimates which compare the quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ with the averages of the corresponding quermassintegrals of the $k$-codimensional sections of $K$ over $G_{n,n-k}$. An example is the…

We obtain a new extension of Rogers-Shephard inequality providing an upper bound for the volume of the sum of two convex bodies $K$ and $L$. We also give lower bounds for the volume of the $k$-th limiting convolution body of two convex…

Here we show that any n-dimensional centrally symmetric convex body K has an n-dimensional perturbation T which is convex and centrally symmetric, such that the isotropic constant of T is universally bounded. T is close to K in the sense…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies…

Let $n$, $q$ and $r$ be positive integers, and let $K_N^n$ be the $n$-skeleton of an $(N-1)$-simplex. We show that for $N$ sufficiently large every embedding of $K_N^n$ in $\mathbb{R}^{2n+1}$ contains a link $L_1\cup\cdots\cup L_r$…

We prove that the Bourgain slicing conjecture and the Kannan-Lov\'asz-Simonovits (KLS) isoperimetric conjecture in $\mathbb{R}^n$ hold true up to a factor of $\sqrt{\log n}$. A new ingredient used in the proof is an improved log-concave…

In this paper we prove that inf_{|z_k| => 1} max_{v=1,...,n^2} |sum_{k=1}^n z_k^v| = sqrt n+O(n^{0.2625+epsilon}). This improves on the bound O(sqrt (n log n)) of Erdos and Renyi. In the special case of $n+1$ being a prime we have…

This article is a survey of recent results on slicing inequalities for convex bodies. The focus is on the setting of arbitrary measures in place of volume.

Let $K$ be a centrally-symmetric convex body in $\mathbb{R}^n$ and let $\|\cdot\|$ be its induced norm on ${\mathbb R}^n$. We show that if $K \supseteq r B_2^n$ then: \[ \sqrt{n} M(K) \leqslant C \sum_{k=1}^{n} \frac{1}{\sqrt{k}}…

We study some geometric properties of the $L_q$-centroid bodies $Z_q(\mu)$ of an isotropic log-concave measure $\mu $ on ${\mathbb R}^n$. For any $2\ls q\ls\sqrt{n}$ and for $\varepsilon \in (\varepsilon_0(q,n),1)$ we determine the inradius…