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![The zone diagram (m1,M2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m_1,M_2)$$\end{document} mentioned in Example 5.5](publication/230672582/figure/fig4/AS:961007816020008@1606133526829/The-zone-diagram-m1-M2documentclass12ptminimal-usepackageamsmath.jpg)
The zone diagram (m1,M2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m_1,M_2)$$\end{document} mentioned in Example 5.5
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![I(2)\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/230672582/figure/fig3/AS:961007816015902@1606133526743/I2documentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![The zone diagram (m1,M2)\documentclass[12pt]{minimal}...](publication/230672582/figure/fig4/AS:961007816020008@1606133526829/The-zone-diagram-m1-M2documentclass12ptminimal-usepackageamsmath_Q320.jpg)
Classical objects in computational geometry are defined by explicit
relations. A few years ago an interesting family of geometric objects defined
by implicit relations was introduced in the pioneering works of T. Asano, J.
Matousek and T. Tokuyama. An important member in this family is a zone diagram,
defined formally as a solution to a fixed point...
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In many applications concerning the comparison of data expressed by
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Citations
A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in $\mathbb{R}^n$, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm must be Euclidean. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possible independent interest, among them results related to coercive bilinear forms (essentially a quantitative convex analytic converse to the celebrated Lax-Milgram theorem from partial differential equations), to infinite-dimensional convex geometry, and to a new class of linear operators (semi-skew operators) which is introduced here.
The (distance) k-sector is a generalization of the concept of bisectors proposed by Asano, Matoušek and Tokuyama. We prove the uniqueness of the 4-sector of two points in the Euclidean plane. Despite the simplicity of the unique 4-sector (which consists of a line and two parabolas), our proof is quite non-trivial. We begin by establishing uniqueness in a small region of the plane, which we show may be perpetually expanded afterward.
Consider a set represented by an inequality. An interesting phenomenon which
occurs in various settings in mathematics is that the interior of this set is
the subset where strict inequality holds, the boundary is the subset where
equality holds, and the closure of the set is the closure of its interior. This
paper discusses this phenomenon assuming the set is a Voronoi cell induced by
given sites (subsets), a geometric object which appears in many fields of
science and technology and has diverse applications. Simple counterexamples
show that the discussed phenomenon does not hold in general, but it is
established in a wide class of cases. More precisely, the setting is a
(possibly infinite dimensional) uniformly convex normed space with arbitrary
positively separated sites. An important ingredient in the proof is a strong
version of the triangle inequality due to Clarkson (1936), an interesting
inequality which has been almost totally forgotten.
The Voronoi diagram is a geometric object which is widely used in many areas.
Recently it has been shown that under mild conditions Voronoi diagrams have a
certain continuity property: small perturbations of the sites yield small
perturbations in the shapes of the corresponding Voronoi cells. However, this
result is based on the assumption that the ambient normed space is uniformly
convex. Unfortunately, simple counterexamples show that if uniform convexity is
removed, then instability can occur. Since Voronoi diagrams in normed spaces
which are not uniformly convex do appear in theory and practice, e.g., in the
plane with the Manhattan (ell_1) distance, it is natural to ask whether the
stability property can be generalized to them, perhaps under additional
assumptions. This paper shows that this is indeed the case assuming the unit
sphere of the space has a certain (non-exotic) structure and the sites satisfy
a certain "general position" condition related to it. The condition on the unit
sphere is that it can be decomposed into at most one "rotund part" and at most
finitely many non-degenerate convex parts. Along the way certain topological
properties of Votonoi cells (e.g., that the induced bisectors are not "fat")
are proved.
