Higher Convexity and Iterated Sum Sets

Convexity and Haar Null Sets

It is shown that for every closed, convex and nowhere dense subset C of a superreflexive Banach space X there exists a Radon probability measure μ on X so that μ(C + x) = 0 for all x ∈ X. In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a clos...

Sum of Squares and Polynomial Convexity

The notion of sos-convexity has recently been proposed as a tractable sufficient condition for convexity of polynomials based on sum of squares decomposition. A multivariate polynomial p(x) = p(x1, . . . , xn) is said to be sos-convex if its Hessian H(x) can be factored as H(x) = M (x) M (x) with a possibly nonsquare polynomial matrix M(x). It turns out that one can reduce the problem of decidi...

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...

Sum-Free Sets and Related Sets

A set A of integers is sum-free if A\(A+A) = ;. Cameron conjectured that the number of sum-free sets A f1; : : : ; Ng is O(2 N=2). As a step towards this conjecture, we prove that the number of sets A f1

Convexity Ranks in Higher Dimension