Characterizations of reduced polytopes finite-dimensional normed spaces

Marek Lassak

Abstract: A convex body $R$ in a normed $d$-dimensional space $M^d$ is called reduced if the $M^d$-thickness $\Delta (K)$ of each convex body $K\subset R$ different from $R$ is smaller than $\Delta (R)$. We present two characterizations of reduced polytopes in $M^d$. One of them is that a convex polytope $P \subset M^d$ is reduced if and only if through every vertex $v$ of $P$ a hyperplane strictly supporting $P$ passes such that the $M^d$-width of $P$ in the perpendicular direction is $\Delta (P)$. Also two characterization of reduced simplices in $M^d$ and a characterization of reduced polygons in $M^2$ are given.

Keywords: reduced body, reduced polytope, normed space, width, thickness, chord

Classification (MSC2000): 52A21, 52B11, 46B20

Full text of the article:

• Compressed PostScript file (505 kilobytes)
• PDF file (172 kilobytes)