arXiv subjects

Gautam Aishwarya

Publications and source records attributed to Gautam Aishwarya.

9 recordsLinked to original sources

arXiv2026

Gaussian Convolution, Internal Energies, and the Kneser--Poulsen Conjecture

We study to what extent the majorisation order between a probability measure and its $1$-Lipschitz image is preserved when both measures undergo Gaussian convolution. We show that the majorisation order is fully preserved in dimensions $n\leq 2$, and obtain dimension-dependent partial preservation in higher dimensions. The perspective taken is that majorisation between densities amounts to comparison through internal energies satisfying a certain pressure condition. Accordingly we introduce a notion of iterated nonnegativity of pressure, closely related to iterated pressures arising in optimal transport, placing the majorisation order within a graded hierarchy of internal energy comparisons. Using the observation that the volume of Euclidean neighbourhoods can be detected from internal energy measurements along the heat flow, we show that our results imply several principal known cases of the Kneser--Poulsen conjecture.

math.PR↗

arXiv2026

Sectional curvature and matrix displacement convexity

We show that the sectional curvature of a Riemannian manifold is nonnegative if, and only if, the entropy tensor is matrix displacement convex. As an application, under the assumption of nonnegative sectional curvature, we obtain intrinsic dimensional strengthenings of the HWI inequality, the evolution variational inequality, and the corresponding Wasserstein contraction along heat flows. Moreover, we show that the intrinsic dimensional Wasserstein contraction inequality in fact characterizes nonnegative sectional curvature.

math.DG↗

arXiv2026

Metric complexity is a Bryant--Tupper diversity

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.

math.MG↗

arXiv2026

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

Given even strongly log-concave random vectors $X_{0}$ and $X_{1}$ in $\mathbb{R}^n$, we show that a natural joint distribution $(X_{0},X_{1})$ satisfies, \begin{equation} e^{ - \frac{1}{n}D ((1-t)X_{0} + t X_{1} \Vert Z)} \geq (1-t) e^{ - \frac{1}{n}D (X_{0} \Vert Z)} + t e^{ - \frac{1}{n}D ( X_{1} \Vert Z)}, \end{equation} where $Z$ is distributed according to the standard Gaussian measure $γ$ on $\mathbb{R}^n$, $t \in [0,1]$, and $D(\cdot \Vert Z)$ is the Gaussian relative entropy. This extends and provides a different viewpoint on the corresponding geometric inequality proved by Eskenazis and Moschidis, namely that \begin{equation} γ\left( (1-t) K_{0} + t K_{1} \right)^{\frac{1}{n}} \geq (1-t) γ(K_{0})^{\frac{1}{n}} + t γ(K_{1})^{\frac{1}{n}}, \end{equation} when $K_{0}, K_{1} \subseteq \mathbb{R}^n$ are origin-symmetric convex bodies. As an application, using Donsker--Varadhan duality, we obtain Gaussian Borell--Brascamp--Lieb inequalities applicable to even log-concave functions, which serve as functional forms of the Eskenazis--Moschidis inequality.

math.PR↗

arXiv2026

New Brunn--Minkowski and functional inequalities via convexity of entropy

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.

math.MG↗

arXiv2025

The Kneser--Poulsen phenomena for entropy

The Kneser--Poulsen conjecture asserts that the volume of a union of balls in Euclidean space cannot be increased by bringing their centres pairwise closer. We prove that its natural information-theoretic counterpart is true. This follows from a complete answer to a question asked in arXiv:2210.12842 about Gaussian convolutions, namely that the Rényi entropy comparisons between a probability measure and its contractive image are preserved when both undergo simultaneous heat flow. An inequality that unifies Costa's result on the concavity of entropy power with the entropic Kneser--Poulsen theorem is also presented.

math.MG↗

arXiv2024

Stability in the Banach isometric conjecture and nearly monochromatic Finsler surfaces

The Banach isometric conjecture asserts that a normed space with all of its $k$-dimensional subspaces isometric, where $k\geq 2$, is Euclidean. The first case of $k=2$ is classical, established by Auerbach, Mazur and Ulam using an elegant topological argument. We refine their method to arrive at a stable version of their result: if all $2$-dimensional subspaces are almost isometric, then the space is almost Euclidean. Furthermore, we show that a $2$-dimensional surface, which is not a torus or a Klein bottle, equipped with a near-monochromatic Finsler metric, is approximately Riemannian. The stability is quantified explicitly using the Banach-Mazur distance.

math.MG↗

arXiv2023

Entropic exercises around the Kneser-Poulsen conjecture

We develop an information-theoretic approach to study the Kneser--Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a $1$-Lipschitz map. We answer this question affirmatively in various cases.

math.MG↗

arXiv2020

Conditional Rényi entropy and the relationships between Rényi capacities

The analogues of Arimoto's definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored.

cs.IT↗