Higher convexity of coamoeba complements

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.

Higher Convexity for Complements of Tropical Varieties

We consider Henriques’ homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over the field of complex Puiseaux series. We conjecture that the complement of a tropical variety has this higher convexity, and prove a weak form of our conje...

Convexity Ranks in Higher Dimension

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...

Substitutes and complements in network flows viewed as discrete convexity

A new light is shed on “substitutes and complements” in the maximum weight circulation problem with reference to the concepts of L-convexity and M-convexity in the theory of discrete convex analysis. This provides us with a deeper understanding of the relationship between convexity and submodularity in combinatorial optimization.