Abstract
Ifg andh are any nonzero functions on the class of convex polytopes then α(F i,F j) =g(F i)/h(F j) is a valuation whose inverse is ω(F i,F j) = (−1)j−i h(F i)/g(F j). This is proved and a smaller class of valuations are characterized: those α(F i,F j) which depend only oni andj and which have inverses of the same form.
Article PDF
Similar content being viewed by others
References
B. Grünbaum,Convex Polytopes, Wiley, London, 1967.
P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes,Math. Proc. Cambridge Philos. Soc. 78 (1975), 247–261.
P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes,Proc. London Math. Soc. (3)35 (1977), 113–135.
G. C. Rota, On the foundations of combinatorial theory, (1). Theory of Möbius functions,Z. Wahrsch. Verw. Gebiete 2 (1964), 340–368.
Author information
Authors and Affiliations
Additional information
Communicated by G.-C. Rota
Rights and permissions
About this article
Cite this article
Sallee, G.T. Invertible relations on polytopes. Discrete Comput Geom 2, 395–399 (1987). https://doi.org/10.1007/BF02187891
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02187891