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A theorem on the average number of subfaces in arrangements and oriented matroids

Authors:	Komei Fukuda Akihisa Tamura Takeshi Tokuyama

Institution:	(1) Graduate School of Systems Management, The University of Tsukuba, 3-29-1 Otsuka, Bunkyo-ku, 112 Tokyo, Japan;(2) Department of Information Sciences, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, 152 Tokyo, Japan;(3) IBM Research, Tokyo Research Laboratory, 5-11 Sanbancho, Chiyoda-ku, 102 Tokyo, Japan

Abstract:	It is known that for simple arrangements in thed-dimensional Euclidean spaceR ^dThe average number ofj-dimensional subfaces of ak-dimensional face is less than $2^{k - j} \left( {\begin{array}{*{20}c} k \\ j \\ \end{array} } \right)$ . In this paper, we show that this is also true for all arrangements inR ^d and for all oriented matroids, and we give combinatorial proofs.

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