A simple proof of a reverse Minkowski theorem for integral lattices

We prove that for any integral lattice $\mathcal{L} \subset \mathbb{R}^n$ (that is, a lattice $\mathcal{L}$ such that the inner product $\langle \mathbf{y}_1,\mathbf{y}_2 \rangle$ is an integer for all $\mathbf{y}_1, \mathbf{y}_2 \in \mathcal{L}$) and any positive integer $k$, $$|\{ \mathbf{y} \in \mathcal{L} \ : \ \|\mathbf{y}\|^2 = k\}| \leq 2 \binom{n+2k-2}{2k-1} \; ,$$ giving a nearly tight reverse Minkowski theorem for integral lattices

Non-contraction of heat flow on Minkowski spaces

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance.Comment: 26 pages; minor revisions, to appear in Arch. Ration. Mech. Ana

A sausage body is a unique solution for a reverse isoperimetric problem

We consider the class of $\lambda$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/\lambda$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/\lambda$ (a sausage body) is a unique volume minimizer among all $\lambda$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.Comment: 1 figur