Concentration inequalities for $s$-concave measures of dilations of Borel sets and applications

几何大偏差型原理与混合测度

Geometric Large-Deviation-Type Principles for Mixed Measures

本文研究了混合测度的几何大偏差型原理，推导了二阶混合测度的积分表示，并证明了凸体在扩张下的渐进行为与包含关系之间的联系。

等重心不等式

Isobarycentric Inequalities

本文证明了给定重心条件下半空间测度最大的等周型不等式，推进了 Henk 和 Pollehn 问题及 Log-Minkowski 不等式的解决，并提出了关于对数凹随机变量截断均值的不等式与猜想。

熵与函数形式的高斯空间维度Brunn-Minkowski不等式

Entropic and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space

本文研究了高斯空间中偶对称log-concave随机向量的熵形式和函数形式的维度Brunn-Minkowski不等式，推导了新的不等式形式并应用Donsker-Varadhan对偶性得到Gaussian Borell-Brascamp-Lieb不等式。

Jensen's inequality in geodesic spaces with lower bounded curvature

Let $(M,d)$ be a separable and complete geodesic space with curvature lower bounded, by $κ\in \mathbb R$, in the sense of Alexandrov. Let $μ$ be a Borel probability measure on $M$, such that $μ\in\mathcal P_2(M)$, and that has at least one barycenter $x^{*}\in M$. We show that for any geodesically $α$-convex function $f:M\to \mathbb R$, for $α\in \mathbb R$, the inequality \[f(x^*)\le \int_M (f -\fracα{2}d^2(x^*,.))\,{\rm d}μ,\] holds provided $f$ is locally Lipschitz at $x^*$ and either positive or in $L^1(μ)$. Our proof relies on the properties of tangent cones at barycenters and on the exis…

Inverse problems in multifractal analysis

Multifractal formalism is designed to describe the distribution at small scales of the elements of $\mathcal M^+_c(\R^d)$, the set of positive, finite and compactly supported Borel measures on $\R^d$. It is valid for such a measure $μ$ when its Hausdorff spectrum is the upper semi-continuous function given by the concave Legendre-Fenchel transform of the free energy function $τ_μ$ associated with $μ$; this is the case for fundamental classes of exact dimensional measures. For any function $τ$ candidate to be the free energy function of some $μ\in \mathcal M^+_c(\R^d)$, we build such a measure,…

热流下的泛函体积积

The functional volume product under heat flow

本文证明了偶函数的泛函体积积在 Fokker--Planck 热流下单调递增，基于对 Ornstein--Uhlenbeck 半群正则化性质的新理解改进了 Borell 反超压缩不等式，并导出了偶函数 Laplace 变换的精确 $L^p$-$L^q$ 不等式。