منابع مشابه
Algebraic boundaries of convex semi-algebraic sets
We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a com...
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A subset B of a closed convex set A is recession-compatible with respect to A if A can be expressed as the Minkowski sum of B and the recession cone of A. We show that if A contains no line, then there exists a recession-compatible subset of A that is minimum with respect to set inclusion. The proof only uses basic facts of convex analysis and does not depend on Zorn’s Lemma. An application of ...
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The Sylvester (d + 2)-points problem deals with the probability S(K) that d + 2 random points taken from a convex compact subset K of Rd are not the vertices of any convex polytope and asks for which sets S(K) is minimal or maximal. While it is known that ellipsoids are the only minimizers of S(K), the problem of the maximum is still open, unless d = 2. In this paper we study generalizations of...
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A convex body K in R has around it a unique circumscribed ellipsoid CE(K) with minimum volume, and within it a unique inscribed ellipsoid IE(K) with maximum volume. The modern theory of these ellipsoids is pioneered by Fritz John in his seminal 1948 paper. This paper has two, related goals. First, we investigate the symmetry properties of a convex body by studying its (affine) automorphism grou...
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A tournament is a complete directed graph. A convex subset is a vertex subset with the property that every two-path beginning and ending inside the convex subset is contained completely within the subset. This paper shows that every nontrivial convex subset is the closure of a subset of vertices of cardinality two. This result leads to algorithms that nd all convex subsets in a tournament in O(...
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ژورنال
عنوان ژورنال: MATHEMATICA SCANDINAVICA
سال: 1968
ISSN: 1903-1807,0025-5521
DOI: 10.7146/math.scand.a-10906