In 1948 Besicovitch proved that an affine image of a regular hexagon may be inscribed into an arbitrary planar convex body. We prove Besicovitch's result using a variational approach based on special approximation by triangles and generalize the Besicovitch theorem to a certain new class of hexagons.
We survey the results on the Banach-Mazur distance between different classes of convex bodies. We hope that our generalization of the Besicovitch theorem may become useful for estimation of the Banach-Mazur distance between planar convex bodies. We examined our special approximation by triangles in some specific cases, and it showed a noticeable improvement in comparison with known general methods.
We also consider the Banach-Mazur distance between a simplex and an arbitrary convex body in the three-dimensional case. Using the idea of an inscribed simplex of maximal volume, we obtain a certain related algebraic optimization problem that provides an upper estimate.
| Identifer | oai:union.ndltd.org:MANITOBA/oai:mspace.lib.umanitoba.ca:1993/9155 |
| Date | 26 September 2012 |
| Creators | Iurchenko, Ivan |
| Contributors | Prymak, Andriy (Mathematics), Gunderson, David (Mathematics) Durocher, Stephane (Computer Science) |
| Source Sets | University of Manitoba Canada |
| Detected Language | English |
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