Celebratio Mathematica — Kuperberg — Complete Bibliography

Complete Bibliography

Works connected to Imre Bárány

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I. Bárány, K. Kuperberg, and T. Zamfirescu: “Total curvature and spiralling shortest paths,” pp. 167–176 in U.S.-Hungarian workshops on discrete geometry and convexity (Budapest, 12–16 July 1999 and Auburn, AL, 21–26 March 2000), published as Discrete Comput. Geom. 30 : 2. Issue edited by G. Fejes Tóth and W. Kuperberg. Springer (New York), August 2003. MR 2007957 Zbl 1043.52004 ArXiv math/0301338 incollection

This paper gives a partial confirmation of a conjecture of Agarwal, Har-Peled, Sharir, and Varadarajan that the total curvature of a shortest path on the boundary of a convex polyhedron in \( \mathbb{R}^3 \) cannot be arbitrarily large. It is shown here that the conjecture holds for a class of polytopes for which the ratio of the radii of the circumscribed and inscribed ball is bounded. On the other hand, an example is constructed to show that the total curvature of a shortest path on the boundary of a convex polyhedron in \( \mathbb{R}^3 \) can exceed \( 2\pi \). Another example shows that the spiralling number of a shortest path on the boundary of a convex polyhedron can be arbitrarily large.

Gábor Fejes Tóth	Related
Tudor I. Zamfirescu	Related
Włodzimierz Kuperberg	Related
Imre Bárány	Related