The illumination bodies of a simplex
The illumination body \(K^\delta\) of a convex body \(K\) in \(\mathbb{R}^n\) is given for \(\delta \geq 0\) by \[ K^\delta : = \biggl\{ x \in \mathbb{R}^n \mid \text{vol}_n \bigl( co [x,K] \bigr) - \text{vol}_n (K) \leq \delta \biggr\}, \] where \(co [x,K]\) is the convex hull of \(x\) and \(K\), and \(\text{vol}_n\) denotes \(n\)-dimensional volume. For sufficiently smooth bodies, \(\text{vol}_n (K^\delta) - \text{vol}_n (K)\) behaves like \(\delta^{2/(n + 1)}\); however, if \(K\) is a polytope, then it is a polynomial in \(\delta\) of degree \(n\). In this paper, the author determines the coefficients of this polynomial, when \(K = S_n\) is an \(n\)-simplex. In fact, \[ \text{vol}_n (S^\delta_n) - \text{vol}_n (S_n) = \sum^n_{k = 1} {n \choose k} {n + k \choose n} {\delta^k \over \bigl( \text{vol}_n (S_n) \bigr)^{k - 1}}. \]
scientific article; zbMATH DE number 881217