204 research outputs found
Mixed volumes and the Bochner method
At the heart of convex geometry lies the observation that the volume of
convex bodies behaves as a polynomial. Many geometric inequalities may be
expressed in terms of the coefficients of this polynomial, called mixed
volumes. Among the deepest results of this theory is the Alexandrov-Fenchel
inequality, which subsumes many known inequalities as special cases. The aim of
this note is to give new proofs of the Alexandrov-Fenchel inequality and of its
matrix counterpart, Alexandrov's inequality for mixed discriminants, that
appear conceptually and technically simpler than earlier proofs and clarify the
underlying structure. Our main observation is that these inequalities can be
reduced by the spectral theorem to certain trivial `Bochner formulas'.Comment: 17 pages; minor correctio
The extremals of Stanley's inequalities for partially ordered sets
Stanley's inequalities for partially ordered sets establish important
log-concavity relations for sequences of linear extensions counts. Their
extremals however, i.e., the equality cases of these inequalities, were until
now poorly understood with even conjectures lacking. In this work, we solve
this problem by providing a complete characterization of the extremals of
Stanley's inequalities. Our proof is based on building a new ``dictionary"
between the combinatorics of partially ordered sets and the geometry of convex
polytopes, which captures their extremal structures.Comment: 54 pages, 7 figures. Added Theorem 1.
Stability of the logarithmic Sobolev inequality via the Föllmer Process
We study the stability and instability of the Gaussian logarithmic Sobolev inequality, in terms of covariance, Wasserstein distance and Fisher information, addressing several open questions in the literature. We first establish an improved logarithmic Sobolev inequality which is at the same time scale invariant and dimension free. As a corollary, we show that if the covariance of the measure is bounded by the identity, one may obtain a sharp and dimension-free stability bound in terms of the Fisher information matrix. We then investigate under what conditions stability estimates control the covariance, and when such control is impossible. For the class of measures whose covariance matrix is dominated by the identity, we obtain optimal dimension-free stability bounds which show that the deficit in the logarithmic Sobolev inequality is minimized by Gaussian measures, under a fixed covariance constraint. On the other hand, we construct examples showing that without the boundedness of the covariance, the inequality is not stable. Finally, we study stability in terms of the Wasserstein distance, and show that even for the class of measures with a bounded covariance matrix, it is hopeless to obtain a dimension-free stability result. The counterexamples provided motivate us to put forth a new notion of stability, in terms of proximity to mixtures of the Gaussian distribution. We prove new estimates (some dimension-free) based on this notion. These estimates are strictly stronger than some of the existing stability results in terms of the Wasserstein metric. Our proof techniques rely heavily on stochastic methods
- …