Related papers: Spectral gaps, symmetries and log-concave perturba…
We investigate properties of measures in infinite dimensional spaces in terms of Poincar\'e inequalities. A Poincar\'e inequality states that the $L^2$ variance of an admissible function is controlled by the homogeneous $H^1$ norm. In the…
We prove a sharp Poincar\'e inequality for subsets $\Omega$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by…
We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case…
In this paper, we consider a certain convolutional Laplacian for metric measure spaces and investigate its potential for the statistical analysis of complex objects. The spectrum of that Laplacian serves as a signature of the space under…
Extending a result of Caffarelli, we provide global Lipschitz changes of variables between compactly supported perturbations of log-concave measures. The result is based on a combination of ideas from optimal transportation theory and a new…
Our main result is an abstract good-$\lambda$ inequality that allows us to consider three self-improving properties related to oscillation estimates in a very general context. The novelty of our approach is that there is one principle…
Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such…
In this short note we observe, on locally symmetric spaces of higher rank, a connection between the growth indicator function introduced by Quint and the modified critical exponent of the Poincar\'e series equipped with the polyhedral…
We study self-improving properties in the scale of Lebesgue spaces of generalized Poincar\'e inequalities in the Euclidean space. We present an abstract setting where oscillations are given by certain operators (e.g., approximations of the…
For any convex set $\Omega \subset {\mathbb R} ^N$, we provide a lower bound for the inverse of the Poincar\'e constant in $W ^ {1, 1}(\Omega)$: it refines an inequality in terms of the diameter due to Acosta-Duran, via the addition of an…
We study the log-concave measures, their characterization via the Pr\'ekopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev…
We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding…
This note is devoted to the proof of convex Sobolev (or generalized Poincar\'{e}) inequalities which interpolate between spectral gap (or Poincar\'{e}) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of…
In this note we establish some rigidity and stability results for Caffarelli's log-concave perturbation theorem. As an application we show that if a 1-log-concave measure has almost the same Poincar\'e constant as the Gaussian measure, then…
These notes are devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension two and higher interpolate between Poincar\'e, logarithmic Sobolev and critical Sobolev (Onofri in dimension…
Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such…
We show that the conjecture of Kannan, Lov\'{a}sz, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form $\rho(|x|_B)dx$ on $\mathbb{R}^n$ and $\rho(t,|x|_B) dx$…
Given a bounded convex open set $\Omega\subseteq \mathbb R^N$, we prove that the Poincar\'e-Sobolev constants $\lambda_{p,q}(\Omega)$ can be bounded from below by the $p$-power of the ratio between the perimeter of $\Omega$ and a suitable…
We consider a "convolution mm-Laplacian" operator on metric-measure spaces and study its spectral properties. The definition is based on averaging over small metric balls. For reasonably nice metric-measure spaces we prove stability of…
We prove, using optimal transport tools, weighted Poincar'e inequalities for log-concave random vectors satisfying some centering conditions. We recover by this way similar results by Klartag and Barthe-Cordero-Erausquin for log-concave…