Amenable groups and measure concentration on spheres

Quasiconvexity at the boundary and concentration effects generated by gradients

We characterize generalized Young measures, the so-called DiPerna-Majda measures which are generated by sequences of gradients. In particular, we precisely describe these measures at the boundary of the domain in the case of the compactification of $\R^{m\times n}$ by the sphere.

球面上和高斯空间中的集中不等式最优常数

Optimal constants in concentration inequalities on the sphere and in the Gauss space

本文研究球面和高斯空间中函数偏离中位数和均值的集中不等式，给出最优常数的界，改进了现有文献中的结果。

Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group $\Iso(\U)$ of isometries of Urysohn's universal complete separable metric space $\mathbb U$, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if $\U$ is replaced with any generalized Urysohn metric space $U$ that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into…

Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the range $N \in (-\infty,1)$, extending the scope from the traditional range $N \in [n,\infty]$. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound, and discover a new case yielding a \emph{single} model space (besides the previously known $N$-sphere and Gaussian measure when $N \in [n,\infty]$): a (pos…

Harmonic Measures on the Sphere via Curvature-Dimension

We show that the family of probability measures on the $n$-dimensional unit sphere, having density proportional to: \[ S^n \ni y \mapsto \frac{1}{|y - x|^{n+α}}, \] satisfies the Curvature-Dimension condition $CD(n-1-\frac{n+α}{4},-α)$, for all $|x| < 1$, $α\geq -n$ and $n\geq 2$. The case $α= 1$ corresponds to the hitting distribution of the sphere by Brownian motion started at $x$ (so-called "harmonic measure" on the sphere). Applications involving isoperimetric, spectral-gap and concentration estimates, as well as potential extensions, are discussed.

Equivariant concentration in topological groups

We prove that, if $G$ is a second-countable topological group with a compatible right-invariant metric $d$ and $(μ_{n})_{n \in \mathbb{N}}$ is a sequence of compactly supported Borel probability measures on $G$ converging to invariance with respect to the mass transportation distance over $d$ and such that $\left(\mathrm{spt} \, μ_{n}, d\!\!\upharpoonright_{\mathrm{spt} \, μ_{n}}, μ_{n}\!\!\upharpoonright_{\mathrm{spt} \, μ_{n}}\right)_{n \in \mathbb{N}}$ concentrates to a fully supported, compact $mm$-space $\left(X,d_{X},μ_{X}\right)$, then $X$ is homeomorphic to a $G$-invariant subspace of …