Related papers: Quantitative stability for the Brunn-Minkowski ine…

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with Schwartzschild solution in the neighborhood of space-like infinity. The result…

It is suggested that the Minkowski vacuum of quantum field theories of a large number of fields N would be gravitationally unstable due to strong vacuum energy fluctuations unless an N dependent sub-Planckian ultraviolet momentum cutoff is…

Elementary proofs of sharp isoperimetric inequalities on a normed space $(\mathbb{R}^n,||\cdot||)$ equipped with a measure $\mu = w(x) dx$ so that $w^p$ is homogeneous are provided, along with a characterization of the corresponding…

We prove stability in the affirmative part of the Busemann-Petty problem on sections of complex convex bodies.

We prove that if a sufficiently regular even log-concave measure satisfies a certain stronger form of the dimensional Brunn-Minkowski conjecture, then it also satisfies the (B)-conjecture. Furthermore, we show that hereditarily convex…

It has long been conjectured that for nonlinear wave equations which satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for…

The theory of coconvex bodies was formalized by A.~Khovanski{\u\i} and V.~Timorin in \cite{KT}. It has fascinating relations with the classical theory of convex bodies, as well as applications to Lorentzian geometry. In a recent preprint…

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $\gamma$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex…

The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…

We consider the problem of the stability (with sharp exponent) of the Lieb--Solovej inequality for symmetric $SU(N)$ coherent states, which was obtained only recently by the authors. Here, we propose an elementary proof of this result,…

Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev inequality in the plane. Then, exploiting the connection…

Minkowski space is shown to be globally stable as a solution to the Einstein--Vlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the "wave zone", and then proving…

A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified…

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with vertex at the origin $o$ and having nonempty interior. The set $A\subset C$ is $C$-coconvex if the volume of $A$ is finite and $A^{\bullet}=C\setminus A$ is a closed convex set.…

We show that if $G$ is a discrete Abelian group and $A \subset G$ has $\|1_A\|_{B(G)} \leq M$ then $A$ is $O(\exp(\pi M))$-stable in the sense of Terry and Wolf.

Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We…

While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in $\mathbb{R}^n$. In doing this they proved a fractional generalization of the…

We adapt Stein's method to isoperimetric and geometric inequalities. The main challenge is the treatment of boundary terms. We address this by using an elliptic PDE with an oblique boundary condition. We apply our geometric formulation of…

For a quiver $Q$ of Dynkin type $\mathbb{A}_n$, we give a set of $n-1$ inequalities which are necessary and sufficient for a linear stability condition (a.k.a. central charge) $Z\colon K_0(Q) \to \mathbb{C}$ to make all indecomposable…

For a domain $\Omega \subset \mathbb{R}^n$ and a small number $\frak{T} > 0$, let \[ \mathcal{E}_0(\Omega) = \lambda_1(\Omega) + {\frak{T}} {\text{tor}}(\Omega) = \inf_{u, w \in H^1_0(\Omega)\setminus \{0\}} \frac{\int |\nabla u|^2}{\int…