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Characterizations of Reduced Polytopes in Finite-Dimensional Normed Spaces

A convex body R in a normed d-dimensional space M is called reduced if the M-thickness ∆(K) of each convex body K ⊂ R different from R is smaller than ∆(R). We present two characterizations of reduced polytopes in M. One of them is that a convex polytope P ⊂ M is reduced if and only if through every vertex v of P a hyperplane strictly supporting P passes such that the M-width of P in the perpendicular direction is ∆(P ). Also two characterization of reduced simplices in M and a characterization of reduced polygons in M are given. MSC 2000: 52A21, 52B11, 46B20