A spiky ball (Q2810744)

Let \(K \subseteq \mathbb{R}^n\) be a convex body with non-empty interior, and let \(i(K)\) be the smallest number of translates of the interior of \(K\) that cover \(K\) (or, equivalently, the smallest number of directions needed for illuminating the whole boundary of \(K\)). It is known that \(i(K) \geq n+1\), with equality e.g.\ if \(K=B^n\) is the Euclidean ball, and conjectured that \(i(K) \leq 2^n\), with equality e.g.\ if \(K=[-1,1]^n\) is a cube; see Chapter VI in [\textit{V. Boltyanski} et al., Excursions into combinatorial geometry. Berlin: Springer (1997; Zbl 0877.52001)].NEWLINENEWLINEThe paper under review shows by a probabilistic argument that there are bodies close to \(B^n\) whose covering numbers grow exponentially in the dimension: given \(1<D <1.116\) and \(n\) sufficiently large, there is a centrally symmetric convex body \(K \subseteq \mathbb{R}^n\) such that \(\frac{1}{D} B^n \subseteq K \subseteq B^n\) and \(i(K) \geq \frac{1}{20}D^n\).