KAIST Discrete Math Seminar

Assume K is a convex body in R^d and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong toX? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? In this lecture I will talk about these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester’s famous four-point problem and from the theory of random polytopes. The second case is when X is the set of lattice points contained in K and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. The methods are, however, very different.

Several classical results in convexity, like the theorems of Caratheodory, Helly, and Tverberg, have colourful versions. In this talk I plan to explain how two methods, the octahedral construction and Sarkaria’s tensor trick, can be used to prove further extensions and generalizations of such colourful theorems.