Higher Convexity and Iterated Second Moment Estimates

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...

Convexity of Moment Polytopes of Algebraic Varieties

We consider the situation of a compact irreducible subvariety of a smooth compact complex variety equipped with a Kähler form preserved by a torus action. We study the image of that subvariety under the moment map of the Kähler form. 1. Main result The moment map of a Hamiltonian T -symplectic/Kähler manifold has been one of the main interests of study for mathematical as well as physical reaso...

Uniform Estimates and Convexity in Capillary Surfaces

We prove uniform convexity of solutions to the capillarity boundary value problem for fixed boundary angle in (0, π/2) and strictly positive capillarity constant provided that the base domain Ω ⊂ R is sufficiently close to a disk in a suitable C-sense.

Convexity Ranks in Higher Dimension

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.