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Affine-inscribed and affine-circumscribed polygons and polytopes

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Abstract

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ3, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ3 (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles.

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Authors
  1. V. V. Makeev
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Additional information

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298.

Translated by B. M. Bekker.

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Makeev, V.V. Affine-inscribed and affine-circumscribed polygons and polytopes. J Math Sci 91, 3518–3525 (1998). https://doi.org/10.1007/BF02434930

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  • DOI: https://doi.org/10.1007/BF02434930

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