Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 47, No. 2, pp. 559-566 (2006)
Characterizations of reduced polytopes finite-dimensional normed spaces
Marek LassakInstitute of Mathematics and Physics, University of Technology, Bydgoszcz 85-796, Poland, e-mail: firstname.lastname@example.org
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
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Electronic version published on: 19 Jan 2007. This page was last modified: 5 Nov 2009.