The illumination bodies of a simplex
type: Item
instance of: scholarly article
title: The illumination bodies of a simplex
zbMATH Open document ID: 0856.52008
DOI: 10.1007/BF02711497
author: Elisabeth M. Werner
published in: Discrete \& Computational Geometry
publication date: 1996-05-22T00:00:00Z
review text: 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, \[ \text{vol}_n (S^\delta_n) - \text{vol}_n (S_n) = \sum^n_{k = 1} {n \choose k} {n + k \choose n} {\delta^k \over \bigl( \text{vol}_n (S_n) \bigr)^{k - 1}}. \]
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 881217
zbMATH Keywords: \(n\)-simplex
zbMATH Keywords: illumination body
zbMATH Keywords: volume
reviewed by: Peter McMullen
MaRDI profile type: Publication
cites work: Q4274007
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description: scientific article; zbMATH DE number 881217