Given x 0 , a point of a convex subset C of a Euclidean space, the two following statements are proven to be equivalent: (i) every convex function f : C → ℝ is upper semi-continuous at x 0 , and (ii) ...

Let $F$ be a quasi-complete locally convex space, $(\Omega, \Sigma, \mu)$ a complete probability space, and $L^1(\mu; F)$ the space of all strongly integrable ...

Convex geometry and combinatorial optimisation form a vibrant nexus of research that bridges theoretical mathematics with practical algorithm design. The study of convex sets and their structural ...

VOUK 1, in a letter in Nature, gave a proof of Cauchy's theorem that the average area of projection of a convex body on a plane equals one-quarter the surface area of the body. The following less ...