Positivity theorems for solidangle polynomials
Abstract
For a lattice polytope P, define A_P(t) as the sum of the solid angles of all the integer points in the dilate tP. Ehrhart and Macdonald proved that A_P(t) is a polynomial in the positive integer variable t. We study the numerator polynomial of the solidangle series sum_{t >= 0} A_P(t) z^t. In particular, we examine nonnegativity of its coefficients, monotonicity and unimodality questions, and study extremal behavior of the sum of solid angles at vertices of simplices. Some of our results extend to more general valuations.
 Publication:

arXiv eprints
 Pub Date:
 June 2009
 arXiv:
 arXiv:0906.4031
 Bibcode:
 2009arXiv0906.4031B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Metric Geometry;
 28A75;
 05A15;
 52C07
 EPrint:
 10 pages