Related papers: Logarithmically-concave moment measures I

In [1], it is established that a convergent observer with an infinite gain margin can be designed for a given nonlinear system when a Riemannian metric showing that the system is differentially detectable (i.e., the Lie derivative of the…

Given a probability measure $\mu$ supported on a convex subset $\Omega$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincar\'e and log-Sobolev type inequalities on $(\Omega,g_0,\mu)$. To this end, we change the…

We study the quantum Riemannian geometry of quantum projective spaces of any dimension. In particular we compute the Riemann and Ricci tensors, using previously introduced quantum metrics and quantum Levi-Civita connections. We show that…

A theorem of E.Lerman and S.Tolman, generalizing a result of T.Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In…

In a recent work, Klartag gave an improved version of Lichnerowicz' spectral gap bound for uniformly log-concave measures, which improves on the classical estimate by taking into account the covariance matrix. We analyze the equality cases…

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By…

We construct Kahler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the…

We prove that a general class of measures, which includes $\log$-concave measures, is $\frac{1}{n}$-concave according to the terminology of Borell, with additional assumptions on the measures or on the sets, such as symmetries. This…

We solve for the SO(3)-invariant Kahler-Einstein metric on $\mathbb{P}^2$ with cone singularities along a smooth conic curve using numerical approach. The numerical results show the sharp range of angles ($(\pi/2,2\pi]$) for the solvability…

Using a Galilean metric approach, based in an embedding of the Euclidean space into a (4+1)-Minkowski space, we analyze a gauge invariant Lagrangian associated with a Riemannian manifold R, with metric g. With a specific choice of the gauge…

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively…

We discuss situations where perturbing a probability measure on $\mathbb{R}^n$ does not deteriorate its Poincar\'e constant by much. A particular example is the symmetric exponential measure in $\mathbb{R}^n$, even log-concave perturbations…

The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from arXiv:2009.05298, local and global…

We study functional inequalities (Poincar\'e, Cheeger, log-Sobolev) for probability measures obtained as perturbations. Several explicit results for general measures as well as log-concave distributions are given.The initial goal of this…

Tian initiated the study of incomplete K\"ahler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2\pi(1-\alpha)$ for $\alpha\in (0, 1)$. In this paper we study…

In this paper, we provide explicit lower bounds with respect to some quantities of interest (parameters of the underlying distribution, dimension, geometrical characteristics of the domain, position of the origin, etc.) on the spectral gap…

We examine the Langevin diffusion confined to a closed, convex domain $D\subset\mathbb{R}^d$, represented as a reflected stochastic differential equation. We introduce a sequence of penalized stochastic differential equations and prove that…

We derive two-sided bounds for moments of linear combinations of coordinates od unconditional log-concave vectors. We also investigate how well moments of such combinations may be approximated by moments of Gaussian random variables.

We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time…

We employ a Markov semigroup approach combined with the $\Gamma$-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincar\'e inequality…