SYMMETRY POINTS OF A CONVEX SET : Basic Properties and Computational Complexity
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چکیده
Given a convex body S ⊂ IRn and a point x ∈ S, let sym(x, S) denote the symmetry value of x in S: sym(x, S) := max{α ≥ 0 : x + α(x− y) ∈ S for every y ∈ S} , which essentially measures how symmetric S is about the point x, and define sym(S) := max x∈S sym(x, S) . We call x∗ a symmetry point of S if x∗ achieves the above supremum. These symmetry measures are all invariant under invertible affine transformation and/or change in norm, and so are of interest in the study of the geometry of convex sets. Furthermore, these measures arise naturally in the complexity theory of interior-point methods. In this study we demonstrate various properties of sym(x, S) such as under operations over S, or as a function of x for a fixed S. Several relations with convex geometry quantities like volume, distance and diameter, cross-ratio distance are proved. Set approximation results are also shown. Furthermore, we provide a characterization of symmetry points x∗. When S is polyhedral and is given as the intersection of halfspaces S = {x ∈ IRn : Ax ≤ b}, then x∗ and sym(S) can be computed by solving m + 1 linear programs of size m× n. We also present an interior-point algorithm that, given an approximate analytic center xa of S, will compute an approximation of sym(S) to any given relative tolerance 2 in no more than 10m ln 10m 2 iterations of Newton’s method.
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تاریخ انتشار 2004