Related papers: An asymptotically sharp form of Ball's inequality …

We establish volume comparison results for balls in Riemannian manifolds with $C^{1,1}$-metrics with a lower bound on the Ricci tensor and for the evolution of spacelike, acausal, causally complete hypersurfaces with an upper bound on the…

We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2(…

Different types of sinc integrals are investigated when the standard sine function is replaced by the generalised $\sin_{p,q}$ in two parameters. A striking generalisation of the improper Dirichlet integral is achieved. A second surprising…

This paper discusses the approximation by de la Vall\'{e}e Poussin means $V_nf$ on the unit sphere. Especially, the lower bound of approximation is studied. As a main result, the strong converse inequality for the means is established.…

We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) a certain random variable on the…

In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first…

We study approximations of smooth convex bodies by random ball-polytopes. We examine the following probability model: let $K\subset{\bf R}^d$ be a convex body such that $K$ slides freely in a ball of radius $R>0$ and has $C^2$ smooth…

In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that $#(K\cap\Z^n)\leq n! \vol(K)+n$, whenever $K\subset\R^n$ is a convex body…

In this work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body and its volume. The conjecture states that among all 2n-dimensional convex bodies with a given volume the euclidean ball has maximal…

Let $S(\rho)=- Tr (\rho \log\rho)$ be the von Neumann entropy of an $N$-dimensional quantum state $\rho$ and $e_2(\rho)$ the second elementary symmetric polynomial of the eigenvalues of $\rho$. We prove the inequality $S(\rho) \le c(N)…

We derive lower estimates for the approximation of the $d$-dimensional Euclidean ball by polytopes with a fixed number of $k$-dimensional faces, $k\in\{0,1,\ldots,d-1\}$. The metrics considered include the intrinsic volume difference and…

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the…

In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves that for every packing of unit spheres, the density in a ball of radius $r$ is at most $\pi/\sqrt{18}+c/r$ for some constant $c$. When $r$ tends to infinity, this gives…

Let $Z$ be an $n$-dimensional Gaussian vector and let $f: \mathbb R^n \to \mathbb R$ be a convex function. We show that: $$\mathbb P \left( f(Z) \leq \mathbb E f(Z) -t\sqrt{ {\rm Var} f(Z)} \right) \leq \exp(-ct^2),$$ for all $t>1$, where…

Let $p\in(1,n)$. If $\Omega$ is a convex domain in $\rn$ whose $p$-capacitary potential function $u$ is $(1-p)/(n-p)$-concave (i.e. $u^{(1-p)/(n-p)}$ is convex), then $\Omega$ is a ball.

In this note, we prove a Trudinger-Moser inequality for conical metric in the unit ball. Precisely, let $\mathbb{B}$ be the unit ball in $\mathbb{R}^N$ $(N\geq 2)$, $p>1$, $g=|x|^{\frac{2p}{N}\beta}(dx_1^2+\cdots+dx_N^2)$ be a conical…

The Blaschke-Santal\'o inequality states that the volume product $|K| \cdot |K^{o}|$ of a symmetric convex body $K \subset \mathbb{R}^n$ is maximized by the standard Euclidean unit-ball. Cordero-Erausquin asked whether the inequality…

We consider the two-dimensional incompressible Navier-Stokes equations with measure initial vorticity. By means of improved Nash inequalities, we establish quantitative estimates for the enstrophy depending on the absolute vorticity decay…

We improve the constant $\frac{\pi}{2}$ in $L^1$-Poincar\'e inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $\sqrt{\frac{\pi}{2}}$. For Hamming cube the sharp constant is not known,…

The two-particle correlation obtained from the quantum state used in the Bell inequality is sinusoidal, but the standard Bell inequality only uses two pairs of settings and not the whole sinusoidal curve. The highest to-date visibility of…