Related papers: Rogers-Shepard Type Inequalities for Sections

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic…

We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any RCD spaces with the Bishop-type inequalities possess only one regular set in not…

We introduce the concept of Calder\'on-Zygmund inequalities on Riemannian manifolds. For $1<p<\infty$, these are inequalities of the form $$ \left\Vert \mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert u\right\Vert…

Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, and assume that X is defined over a function field admitting K as a completion. Let further m be a positive measure on X…

In this paper, we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called $RCD^*(K,N)$ spaces) with \emph{non-empty} one dimensional regular sets. In particular, we prove…

The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called $m$th-order…

The difference body operator enjoys different characterization results relying on its basic properties such as continuity, SL(n)-covariance, Minkowski valuation or symmetric image. The Rogers-Shephard and the Brunn-Minkowski inequalities…

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…

We show that for any isotropic log-concave probability measure $\mu$ on $\mathbb R^n$, for every $\varepsilon > 0$, every $1 \leq k \leq \sqrt{n}$ and any $E \in G_{n,k}$ there exists $F \in G_{n,k}$ with $d(E,F) < \varepsilon$ and…

Let $ m, n $ be integers such that $ \frac{n}{2} > m \geq 1 $ and let $ (M, g) $ be a closed $ n-$dimensional Riemannian manifold. We prove there exists some $ B \in \mathbb{R} $ depending only on $ (M, g) $, $ m $, and $ n $ such that for…

The Hardy-Littlewood inequalities for $m$-linear forms on $\ell_{p}$ spaces are stated for $p>m$. In this paper, among other results, we investigate similar results for $1\leq p\leq m.$ Let $\mathbb{K}$ be $% \mathbb{R}$ or $\mathbb{C}$ and…

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits…

We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio…

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…

Borell's inequality states the existence of a positive absolute constant $C>0$ such that for every $1\leq p\leq q$ $$ \left(\mathbb E|\langle X, e_n\rangle|^p\right)^\frac{1}{p}\leq\left(\mathbb E|\langle X,…

We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube in R^n whose density takes the form exp(-H) where the function H is assumed to be…

The inequality of Berwald is a reverse-H\"older like inequality for the $p$th average, $p\in (-1,\infty),$ of a non-negative, concave function over a convex body in $\mathbb{R}^n.$ We prove Berwald's inequality for averages of functions…

We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood…

We consider a homogeneous space $X=(X,d,m) $ of dimension $\nu\geq1$ and a local regular Dirichlet form in $L^{2}(X,m) .$ We prove that if a Poincar\'{e} inequality holds on every pseudo-ball $B(x,R) $ of $X$, then an Harnack's inequality…

We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $\gamma$ on $\mathbb{R}^n$, stating…