Gautam Aishwarya

Gautam Aishwarya, Dongbin Li, Mokshay Madiman et al.

The metric complexity (sometimes called Leinster--Cobbold maximum diversity) of a compact metric space is a recently introduced isometry-invariant of compact metric spaces which generalizes the notion of cardinality, and can be thought of as a metric-sensitive analogue of maximum entropy. On the other hand, the notion of diversity introduced by Bryant and Tupper is an assignment of a real number to every finite subset of a fixed set, which generalizes the notion of a metric. We establish a connection between these concepts by showing that the former quantity naturally produces an example of the latter. Moreover, in contrast to several examples in the literature, the diversity that arises from metric complexity is Minkowski-superadditive for compact subsets of the real line.

Gautam Aishwarya, Liran Rotem

We study the connection between the concavity properties of a measure $ν$ and the convexity properties of the associated relative entropy $D(\cdot \Vert ν)$ along optimal transport. As a corollary we prove a new dimensional Brunn--Minkowski inequality for centered star-shaped bodies, when the measure $ν$ is log-concave with a p-homogeneous potential (such as the Gaussian measure). Our method allows us to go beyond the usual convexity assumption on the sets that is fundamentally essential for the standard differential-geometric technique in this area. We then take a finer look at the convexity properties of the Gaussian relative entropy, which yields new functional inequalities. First we obtain curvature and dimensional reinforcements to Otto--Villani's HWI inequality in Gauss space, when restricted to even strongly log-concave measures. As corollaries, we obtain improved versions of Gross' Logarithmic Sobolev inequality and Talagrand's transportation cost inequality in this setting.