Related papers: On the reverse Loomis-Whitney inequality

In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|\theta^{\perp}|…

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 $\es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ \sum_{n=2}^{\infty}a_nz^n$. Let $F$ be the inverse of the function $f\in\es$ with the series expansion %in a disk…

For every involution $\mathbf{w}$ of the symmetric group $S_n$ we establish, in terms ofa special canonical quotient of the dominant Verma module associated with $\mathbf{w}$, an effective criterion, which allows us to verify whether the…

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy.…

The classical Loomis-Whitney inequality and the uniform cover inequality of Bollob\'{a}s and Thomason provide lower bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further…

We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be…

A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the…

A certain inequality conjectured by Vershynin is studied. It is proved that for any $n$-dimensional symmetric convex body $K$ with inradius $w$ and $\gamma_{n}(K) \leq 1/2$ there is $\gamma_{n}(sK) \leq (2s)^{w^{2}/4}\gamma_{n}(K)$ for any…

Determination of polymer molecular weight distribution (MWD) from rheological measurements is desirable due to the ease and low cost of rheometry compared to other methods such as gel permeation chromatography. However, relating MWD to…

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures…

For $n\geq 2$, let $\Gamma\subset \mathrm{SU}((n,1),\mathcal{O}_{K})$ be a torsion-free, finite-index subgroup, where $\mathcal{O}_K$ denotes the ring of integers of a totally imaginary number field $K$ of degree $2$. Let $\mathbb{B}^n$…

The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose…

We use the characterization of the case of equality in Barthe's Geometric Reverse Brascamp-Lieb inequality to characterize equality in Liakopoulos's volume estimate in terms of sections by certain lower-dimensional linear subspaces.

We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space…

For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull…

For a convex body $K \subset {\mathbb R}^n$, let $K^z = \{y\in{\mathbb R}^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of $K$ with respect to the center of polarity $z \in {\mathbb R}^n$. The goal of this…

Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating the curvature of $\ell K$,…

This note concerns Loomis-Whitney inequalities in Heisenberg groups $\mathbb{H}^n$: $$|K| \lesssim \prod_{j=1}^{2n}|\pi_j(K)|^{\frac{n+1}{n(2n+1)}}, \qquad K \subset \mathbb{H}^n.$$ Here $\pi_{j}$, $j=1,\ldots,2n$, are the vertical…

Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such…