# Affine cross-polytopes inscribed in a convex body

- 21 Downloads

Let *X* be an affine cross-polytope, i.e., the convex hull of n segments *A* _{1} *B* _{1},…, *A* _{ n } *B* _{ n } in \( {\mathbb{R}^n} \) that have a common midpoint *O* and do not lie in a hyperplane. The affine flag *F*(*X*) of *X* is the chain *O* ∈ *L* _{1} ⊂⋯ ⊂ *L* _{ n } = \( {\mathbb{R}^n} \), where *L* _{ k } is the *k*-dimensional affine hull of the segments *A* _{1} *B* _{1},…, *A* _{ k } *B* _{ k }, *k* ≤ *n*. It is proved that each convex body *K* ⊂ \( {\mathbb{R}^n} \) is circumscribed about an affine cross-polytope *X* such that the flag *F*(*X*) satisfies the following condition for each *k* ∈{2,…, *n*}:the (*k*−1)-planes of support at *A* _{ k } and *B* _{ k } to the body *L* _{ k } ∩ *K* in the *k*-plane *L* _{ k } are parallel to *L* _{ k }−1.Each such *X* has volume at least *V*(*K*)/2^{n(n−1)/2}. Bibliography: 5 titles.

## Keywords

Russia Hull Convex Hull Convex Body Affine Hull## References

- 1.V. L. Dol'nikov, “Transversals of families of sets in \( {\mathbb{R}^n} \) and the relation between Helly's and Borsuk's theorems,”
*Mat. Sb.*,**184**, No. 5, 111–132 (1993).Google Scholar - 2.E. Fadell and S. Husseini, “An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems,”
*Ergodic Theory Dynam. Syst.*,**8**, 73–85 (1988).MathSciNetCrossRefGoogle Scholar - 3.K. Leichtweiss,
*Konvexe Mengen*, Springer, Berlin (1979).Google Scholar - 4.R. T. Živaljević and S. T. Vrećica, “An extension of the ham sandwich theorem,”
*Bull. London Math. Soc.*,**22**, 183–186 (1990).MathSciNetMATHCrossRefGoogle Scholar - 5.V. Soltan, “Affine diameters of convex bodies-a survey,”
*Expo. Math.*,**23**, 47–63 (2005).MathSciNetMATHGoogle Scholar