Dilation type inequalities for strongly-convex sets in weighted Riemannian manifolds (Q2071821)

The main purpose of this paper is to establish the sharp dilation type inequalities under the curvature condition \(\mathrm{Ric}\geq K\) for some \(K\in \mathbb{R}\) for a weighted Riemannian manifold. The author introduces the notion of \(\epsilon\)-dilation profile of a geodesically-convex \(n\)-dimensional Riemannian manifold \((M,g,m)\) with a weighted measure \(m\) satisfying \(m(M)=1\). Some functional inequalities related to various entropies have also been investigated from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature