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We prove that any simple polytope (and some non-simple polytopes) in R 3 admits an inscribed regular octahedron.

For convex partitions of a convex body B we prove that we can put a homo-thetic copy of B into each set of the partition so that the sum of homothety coefficients is ≥ 1. In the plane the partition may be arbitrary, while in higher dimensions we need certain restrictions on the partition.

Consider a polygon P and all neighboring circles (circles going through three consecutive vertices of P). We say that a neighboring circle is extremal if it is empty (no vertices of P inside) or full (no vertices of P outside). It is well known that for any convex polygon there exist at least two empty and at least two full circles, i.e. at least four… (More)

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