The polar dual of a convex polyhedral set in hyperbolic space
type: Item
instance of: scholarly article
title: The polar dual of a convex polyhedral set in hyperbolic space
zbMATH Open document ID: 0860.52007
author: Ruth Charney
author: Michael W. Davis
published in: Michigan Mathematical Journal
publication date: 1997-04-15T00:00:00Z
review text: Let \(V\) be a finite-dimensional vector space. A subset \(X\) of \(V\) is a convex polyhedral set (polyhedral cone \(C\)) if it is defined by a finite number of affine (linear) inequalities. Let \(X\) be a convex polyhedral set in a space of constant curvature and denote by \(P(X)\) its polar dual. A piecewise spherical cell complex, with its intrinsic metric, is ``large'' if there is a unique geodesic between any two points of distance less than \(\pi\). A polyhedron of piecewise constant curvature has curvature bounded from above if and only if the link of each point is large. Let \(S_\infty(X)\) denote the points at infinity of \(X\). For any \(y\in S_\infty(X)\), let \(F_y\) be the unique face of \(C(X)\) that contains \(y\) in its relative interior. The point \(y\) is called a ``cusp'' point of \(X\) if \(F_y\) is lightlike. If \(y\) is a cusp point of \(X\), then put \(P_y= P_{F_y}= F_y^\perp\cap P(X)\). The main result is: Suppose \(X\) is a hyperbolic convex polyhedral set of dimension \(n\). Then, (1) its polar dual \(P(X)\) is large, and (2) if \(\gamma\) is any closed local geodesic of length \(2\pi\), then \(\gamma\) must lie in the subcomplex \(P_y\) for some cusp point \(y\) of \(X\). The last sections relate the main result to the following conjecture: For a \(p\)-dimensional spherical cell \(\sigma\), let \(a(\sigma)\) denote the \(p\)-dimensional volume suitable normalized and let \(a^*(\sigma)= a(\sigma^*)\) where \(\sigma^*\) is the dual cell to \(\sigma\). Given a finite, piecewise spherical cell complex \(K\), consider the quantity \(\kappa(K)= 1+\sum_{\sigma^{-1}} (\dim \sigma)+1\cdot a^*(\sigma)\), where the summation is over all cells \(\sigma\) of \(K\). Suppose that \(K\) is a large piecewise spherical structure on the \((2m-1)\)-sphere, then \((-1)^m \kappa(K)\geq 0\). The sign of \(\kappa(K)\) is correct when \(K\) is the polar dual of a hyperbolic polytope.
reviewed by: L. A. Santaló
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 52A55
Mathematics Subject Classification ID: 53A35
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 881136
zbMATH Keywords: convex polyhedral set
zbMATH Keywords: geodesic
zbMATH Keywords: polar dual
zbMATH Keywords: cusp point
zbMATH Keywords: hyperbolic polytope
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005308
OpenAlex ID: W1540752843
DOI: 10.1307/MMJ/1029005308
Recommended article: Polar Duals of Convex Bodies
Recommended article: Dual representations of convex bodies and their polars
Recommended article: On hyperbolic sets of maxes in dual polar spaces
Recommended article: On hyperovals of polar spaces
Recommended article: Dimensional dual hyperovals in classical polar spaces
Recommended article: Convex hulls in the hyperbolic space
Recommended article: The polar of convex lattice sets
Recommended article: Polar Means of Convex Bodies and a Dual to the Brunn-Minkowski Theorem
Recommended article: Polarized and homogeneous embeddings of dual polar spaces
Recommended article: Hyperplanes and projective embeddings of dual polar spaces
description: scientific article; zbMATH DE number 881136