On the theory of coconvex bodies
This article studies inequalities related to a coconvex body \(A\). By definition \(A = C \setminus (\Delta \cup \{0\})\) where \(C\) is a convex cone with apex \(\{0\}\) and \(\Delta \subset C\) a closed convex subset such that \(A\) is bounded. The main motivation of the authors to study coconvex bodies comes from the local study of singularities of polynomials (in guise of Newton diagrams). One of the main results is a coconvex Alexandrov-Fenchel inequality, which gives in turn coconvex versions of the Brunn--Minkowski and the Minkowski inequalities. This generalises work by \textit{F. Fillastre} [Geom. Funct. Anal. 23, No. 1, 295--333 (2013; Zbl 1271.52009)] and \textit{K. Kaveh} and \textit{A. G. Kohanskiĭ} [Proc. Steklov Inst. Math. 286, 268--284 (2014; Zbl 1315.13013)]. Moreover, in the case of a coconvex integral polytope, they give a formula for the generating function of integer points in it, which generalises the formula for convex polytopes by \textit{M. Brion} [Ann. Sci. Éc. Norm. Supér. 21, No. 4, 653--663 (1988; Zbl 0667.52011)]. Both results are proven by using virtual convex polytopes, i.e.\ formal differences of polytopes, and hence passing to the analoguous statements in convex geometry. The authors show that the indicator function of a coconvex body \(A = C \setminus (\Delta \cup \{0\})\) can be expressed using the convolution product of (inverse) indicator functions of polytopes derived from \(C\) and \(\Delta\). Actually, here it becomes important to remove the apex \(\{0\}\).
scientific article