Inscribed balls and the Lyusternik-Shnirel'man category (Q1375027)

The problem under study is to count the number of balls \(N(K)\) inscribed in a compact set \(K\subset \mathbb R^n\). For compacts with \(C^1\)-smooth boundary, the author proves the inequality \(N(K)\geq \text{cat} K\). Here \(\text{cat} K\) denotes the Lyusternik-Shnirel'man category of the compact set \(K\). For proving this inequality, the author constructs some special function \(f\: K\to \mathbb R\) and studies its critical points.