Combinatorial identities for polyhedral cones

Schneider, R.

doi:10.1090/spmj/1489

Combinatorial identities for polyhedral cones
HTML articles powered by AMS MathViewer

by R. Schneider

St. Petersburg Math. J. 29 (2018), 209-221

DOI: https://doi.org/10.1090/spmj/1489

Published electronically: December 27, 2017

PDF | Request permission

Abstract:

Some known relations for convex polyhedral cones, involving angles or conical intrinsic volumes, are superficially of a metric character, but have indeed a purely combinatorial core. This fact is strengthened in some cases, with implications for valuations on polyhedral cones, and is worked out in the case of the extended Klivans–Swartz formula.

References

Dennis Amelunxen and Martin Lotz, Intrinsic volumes of polyhedral cones: a combinatorial perspective, Discrete Comput. Geom. 58 (2017), no. 2, 371–409. MR 3679941, DOI 10.1007/s00454-017-9904-9
Beifang Chen, The Gram-Sommerville and Gauss-Bonnet theorems and combinatorial geometric measures for noncompact polyhedra, Adv. Math. 91 (1992), no. 2, 269–291. MR 1149626, DOI 10.1016/0001-8708(92)90019-H
H. Groemer, On the extension of additive functionals on classes of convex sets, Pacific J. Math. 75 (1978), no. 2, 397–410. MR 513905
Branko Grünbaum, Grassmann angles of convex polytopes, Acta Math. 121 (1968), 293–302. MR 238179, DOI 10.1007/BF02391916
Branko Grünbaum, Convex polytopes, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR 1976856, DOI 10.1007/978-1-4613-0019-9
H. Hadwiger, Eulers Charakteristik und kombinatorische Geometrie, J. Reine Angew. Math. 194 (1955), 101–110 (German). MR 73221, DOI 10.1515/crll.1955.194.101
Daniel Hug and Rolf Schneider, Random conical tessellations, Discrete Comput. Geom. 56 (2016), no. 2, 395–426. MR 3530973, DOI 10.1007/s00454-016-9788-0
Zakhar Kabluchko, Vladislav Vysotsky, and Dmitry Zaporozhets, Convex hulls of random walks, hyperplane arrangements, and Weyl chambers, Geom. Funct. Anal. 27 (2017), no. 4, 880–918. MR 3678504, DOI 10.1007/s00039-017-0415-x
Zakhar Kabluchko, Günter Last, and Dmitry Zaporozhets, Inclusion-exclusion principles for convex hulls and the Euler relation, Discrete Comput. Geom. 58 (2017), no. 2, 417–434. MR 3679943, DOI 10.1007/s00454-017-9880-0
Caroline J. Klivans and Ed Swartz, Projection volumes of hyperplane arrangements, Discrete Comput. Geom. 46 (2011), no. 3, 417–426. MR 2826960, DOI 10.1007/s00454-011-9363-7
Hanfried Lenz, Mengenalgebra und Eulersche Charakteristik, Abh. Math. Sem. Univ. Hamburg 34 (1969/70), 135–147 (German). MR 268777, DOI 10.1007/BF02992458
P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 2, 247–261. MR 394436, DOI 10.1017/S0305004100051665
Peter McMullen, Angle-sum relations for polyhedral sets, Mathematika 33 (1986), no. 2, 173–188 (1987). MR 882490, DOI 10.1112/S0025579300011165
—, Valuations and dissections, Handbook of convex geometry, vol. B, North-Holland, Amsterdam, 1993, pp. 933–988. MR (85f:52018)
Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326, DOI 10.1007/978-3-540-78859-1
G. C. Shephard, An elementary proof of Gram’s theorem for convex polytopes, Canadian J. Math. 19 (1967), 1214–1217. MR 225228, DOI 10.4153/CJM-1967-110-7
Richard P. Stanley, An introduction to hyperplane arrangements, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 389–496. MR 2383131, DOI 10.1090/pcms/013/08
Richard P. Stanley, Enumerative combinatorics. Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012. MR 2868112
Thomas Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc. 1 (1975), no. issue 1, 154, vii+102. MR 357135, DOI 10.1090/memo/0154