Tetrahedra are not reduced
A convex body (i.e., a compact convex set with interior points in \(\mathbb R^n\), \(n\geq 2\)) which does not properly contain a convex body with the same minimal width is said to be reduced. It is known that every regular \(m\)-gon in \(\mathbb R^2\) with odd vertex number \(m\) is reduced. Being motivated by the problem `Do there exist reduced \(n\)-polytopes for \(n\geq 3\)?' posed by \textit{M. Lassak} in [T. Bisztriczky, P. McMullen, R. Schneider, and A. Weiss (eds.), Polytopes: abstract, convex and computational (Dordrecht: Kluwer Academic Publishers) (1994; Zbl 0797.00016)], the authors prove that there is no reduced (nondegenerate) tetrahedron in \(\mathbb R^3\). In the general case, Lassak's problem is still open.
scientific article; zbMATH DE number 1878896