Convexity, critical points, and connectivity radius

@article{Katz2019ConvexityCP,
title={Convexity, critical points, and connectivity radius},
author={Mikhail G. Katz},
journal={Proceedings of the American Mathematical Society},
year={2019}
}

We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the critical points of the distance function in the sense of Grove--Shiohama--Gromov--Cheeger.

We investigate those spherical point sets which, relative to the Hausdorff metric, give local minima of the diameter function, and obtain estimates which, in principle, justify computer-generated… Expand

We give an upper bound for the Betti numbers of a compact Riemannian manifold in terms of its diameter and the lower bound of the sectional curvatures. This estimate in particular shows that most… Expand

A basic problem in Riemannian geometry is the study of relations between the topological structure and the Riemannian structure of a complete, connected Riemannian manifold M of dimension n > 2. By a… Expand