Characterizations of reduced polytopes in finite-dimensional normed spaces
Let \(M^d\) be a \(d\)-dimensional normed real space. A strip \(S\) in \(M^d\) is the convex hull of two (distinct) parallel hyperplanes \(H_1\) and \(H_2\), its direction \(m\) is a (euclidean) perpendicular to the \(H_j\), and its width \(w(S)\) is the \(M^d\)-distance between \(H_1\) and \(H_2\). If \(H_1\) and \(H_2\) support a convex body \(C\), then \(S\) is a \(C\)-strip; if its direction is \(m\), then it is denoted by \(S(C,m)\), and \(w(S(C,m))\) is called the \(M^d\)-width of \(C\) in this direction. The number \(\Delta(C) = \min_mw(C,m)\) is the \(M^d\)-thickness of \(C\). If \(c_j \in H_j \cap C\) for \(j = 1,2\) and \(\| c_1 - c_2\| = \Delta(C)\), then \(c_1c_2\) is a thickness chord of \(C\). Denote by \(\Gamma(C,m)\) the \(M^d\)-length of the longest chord of \(C\) in direction \(m\), and set \(\Gamma(C) = \min_m\Gamma(C,m)\). Finally, a convex body \(R\) is called reduced if \(\Delta(K) < \Delta(R)\) whenever \(K \subset R\). The author proves here two equivalent conditions for a polytope \(P\) to be reduced. First, through each vertex \(v\) of \(P\) passes a strip of \(M^d\)-width \(\Delta(P)\) whose bounding hyperplane through \(v\) strictly supports \(P\) (that is, in \(v\) alone). Second, for each vertex \(V\) of \(P\) there is a direction such that the longest chord of \(P\) in this direction is unique, has \(M^d\)-length \(\Gamma(P)\) and \(v\) as its end-point. Also given are two characterizations of reduced simplices in \(M^d\), and one of reduced polygons in \(M^2\).
scientific article; zbMATH DE number 5117822