On the sum of external angles of a convex polyhedron

Kouki Taniyama

KNOTS in WASHINGTON XV (2nd Japan-USA Workshop in Knot Theory)
January 10-15, 2003
George Washington University and Johns Hopkins University
Washington, DC and Baltimore, MD, USA

Organizers
Kazuaki Kobayashi, Jozef H. Przytycki, Yongwu Rong, Shin-ichi Suzuki, Kouki Taniyama, Tatsuya Tsukamoto, Akira Yasuhara

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On the sum of external angles of a convex polyhedron
by
Kouki Taniyama
Waseda University
Coauthors: Kazuhiro Ichihara (Nara Women's University)

Let P be a convex polyhedron in an n-dimensional Euclidean space Eⁿ. For a codimension one subspace V in Eⁿ, let P_V denote the image of P under the orthogonal projection to V. Note that this P_V is a convex polyhedron in V. Let l_V be the number of (n-2)-cells in P_V. Let l be the average of l_V where V varies over all codimension one subspaces in Eⁿ. Then we show that the sum of all external angles of P is equal to l\pi.

Date received: December 25, 2002