Related papers: A nonabelian Brunn-Minkowski inequality

We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's…

Yuan and Leng (2007) gave a generalization of Ky Fan's determinantal inequality, which is a celebrated refinement of the fundamental Brunn-Minkowski inequality $(\det (A+B))^{1/n} \ge (\det A)^{1/n} +(\det B)^{1/n}$, where $A$ and $B$ are…

The Ehrhard-Borell inequality is a far-reaching refinement of the classical Brunn-Minkowski inequality that captures the sharp convexity and isoperimetric properties of Gaussian measures. Unlike in the classical Brunn-Minkowski theory, the…

We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Marde\v{s}i\'{c} and Prasolov…

In this paper, we study the $L_p$ dual Minkowski problem for all $q, p \in \mathbb{R}$ from an algebraic perspective. We establish the existence of solutions for group-invariant convex bodies (not necessarily origin-symmetric), thereby…

The classical Minkowski inequality implies that the volume of a bounded convex domain is controlled from above by the integral of the mean curvature of its boundary. In this note, we establish an analogous inequality without the convexity…

Let $\H= < a,b | a[a,b]=[a,b]a \wedge b[a,b]=[a,b]b>$ be the discrete Heisenberg group, equipped with the left-invariant word metric $d_W(\cdot,\cdot)$ associated to the generating set ${a,b,a^{-1},b^{-1}}$. Letting $B_n= {x\in \H:…

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite…

We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a…

We discuss some techniques related to equivariant compactifications of uniform spaces and amenability of topological groups. In particular, we give a new proof of a recent result by Glasner and Weiss describing the universal minimal flow of…

In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies when the group $G\subset O(n)$ has no nonzero fixed…

If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then…

In this review paper, we survey Hardy type inequalities from the point of view of Folland and Stein's homogeneous groups. Particular attention is paid to Hardy type inequalities on stratified groups which give a special class of homogeneous…

Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining…

We establish a nonlinear generalisation of the classical Brascamp-Lieb inequality in the case where the Lebesgue exponents lie in the interior of the finiteness polytope. As a corollary we show that the best constant in Young's convolution…

By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the…

Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski…

We establish a monotonicity property of the deficit associated with the local log-Brunn-Minkowski inequality (LLBM) under addition of line segments. As a corollary, if the LLBM holds for a convex body K, then it also holds for K+Z for any…

Given two non-empty subsets $A$ and $B$ of the hyperbolic plane $\mathbb{H}^2$, we define their horocyclic Minkowski sum with parameter $\lambda=1/2$ as the set $[A:B]_{1/2} \subseteq \mathbb{H}^2$ of all midpoints of horocycle curves…

A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A…