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Tetrahedra are not reduced. (English) Zbl 1018.52006

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 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.

MSC:

52B10 Three-dimensional polytopes
52A15 Convex sets in \(3\) dimensions (including convex surfaces)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52B11 \(n\)-dimensional polytopes

Citations:

Zbl 0797.00016
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Full Text: DOI

References:

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