Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

. For a given convex body K in with C 2 boundary, let P c n be the circumscribed polytope of minimal volume with at most n edges, and let P i n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity...

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Bibliographic Details
Published inMonatshefte für Mathematik Vol. 153; no. 1; pp. 25 - 48
Main Authors Böröczky, Károly J., Gomis, Salvador S., Tick, Péter
Format Journal Article
LanguageEnglish
Published Vienna Springer-Verlag 01.01.2008
Subjects
Mathematics
Mathematics and Statistics
2000 Mathematics Subject Classification: 52A27, 52A40
Key words: Polytopal approximation, extremal problems
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ISSN0026-9255
1436-5081
DOI10.1007/s00605-007-0496-y

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Summary:. For a given convex body K in with C 2 boundary, let P c n be the circumscribed polytope of minimal volume with at most n edges, and let P i n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of P c n and P i n are asymptotically regular triangles and squares, respectively, in a suitable sense.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-007-0496-y