The holomorphic closure dimension is one of the possible biholomorphic invariant that gives some information on how much of the complex ambient structure a subset of a complex manifold inherits. I will review a series of results by Rasul, Rasul and Janusz, and Janusz and myself that discuss the behaviour of the holomorphic dimension in a real analytic set on the one hand and in semialgebraic sets on the other hand.

We discuss a (new) connection between algebraic geometry/representation theory and convex geometry. We explain a basic construction which associates convex bodies to semigroups of integral points. We see how this gives rise to convex bodies associated to algebraic varieties encoding information about their geometry. This far generalizes the notion of Newton polytope of a toric variety. As an application, we give a formula for the number of solutions of an algebraic system of equations (equivalently self-intersection of a divisor/linear system) on any variety, in terms of volumes of these bodies. This has many interesting applications in algebraic geometry, in particular theory of linear systems. We will see how several convex polytopes naturally appearing in representation theory (of Lie groups) are special cases of this geometric construction. The origin of this approach goes back to influential work of A. Okounkov on multiplicities of representations.