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

Abstract

We study the persistence under heat flow of internal energy comparisons induced by $1$-Lipschitz maps. We define a new hierarchy of internal energies based on iterative nonnegativity of the associated pressure law, and show that Gaussian marginalisation leads to hierarchical ascent. This mechanism, combined with a stronger dimension-free result for continuously contracting homotopies, allows us to prove that persistence holds for tiers in the hierarchy depending on the ambient dimension, including all convex energy densities in the two-dimensional case. We then connect these results to geometry, specifically to the volume of tubes, showing that several principal known cases of the Kneser--Poulsen conjecture follow from our results. Partly motivated by this connection, we conjecture that persistence holds for all convex energy densities in all dimensions. We also discuss a characterisation of isotropic Gaussians arising naturally in this framework.