Convexity ranks in higher dimensions

Convexity Ranks in Higher Dimension

Convexity, complexity, and high dimensions

We discuss metric, algorithmic and geometric issues related to broadly understood complexity of high dimensional convex sets. The specific topics we bring up include metric entropy and its duality, derandomization of constructions of normed spaces or of convex bodies, and different fundamental questions related to geometric diversity of such bodies, as measured by various isomorphic (as opposed...

Inequalities and Higher Order Convexity

We study the following problem: given n real arguments a1, ..., an and n real weights w1, ..., wn, under what conditions does the inequality w1f(a1) + w2f(a2) + · · ·+ wnf(an) ≥ 0 hold for all functions f satisfying f (k) ≥ 0 for some given integer k? Using simple combinatorial techniques, we can prove many generalizations of theorems ranging from the Fuchs inequality to the criterion for Schur...

Higher Convexity of Coamoeba Complements

We show that the complement of the closure of the coamoeba of a variety of codimension k+1 is k-convex, in the sense of Gromov and Henriques. This generalizes a result of Nisse for hypersurface coamoebas. We use this to show that the complement of the nonarchimedean coamoeba of a variety of codimension k+1 is k-convex.

Restricted-Orientation Convexity in Higher-Dimensional Spaces

A restricted-oriented convex set is a set whose intersection with any line from a fixed set of orientations is either empty or connected. This notion generalizes both orthogonal convexity and normal convexity. The aim of this paper is to establish a mathematical foundation for the theory of restricted-oriented convex sets in higher-dimensional spaces.