An Explicit Constant for Sullivan's Convex Hull Theorem

Quasiconformal Lipschitz Maps, Sullivan's Convex Hull Theorem and Brennan's Conjecture

A cone theoretic Krein-Milman theorem in semitopological cones

In this paper, a Krein-Milman  type theorem in $T_0$ semitopological cone is proved,  in general. In fact, it is shown that in any locally convex $T_0$ semitopological cone, every convex compact saturated subset is the compact saturated convex hull of its extreme points, which improves the results of Larrecq.

Convexifying the set of matrices of bounded rank: applications to the quasiconvexification and convexification of the rank function

We provide an explicit description of the convex hull of the set of matrices of bounded rank, restricted to balls for the spectral norm. As applications, we deduce two relaxed forms of the rank function restricted to balls for the spectral norm: one is the quasiconvex hull of this rank function, another one is the convex hull of the rank function, thus retrieving Fazel's theorem (2002). Key-wor...

Approximating the Colorful Carathéodory Theorem

Let P1, . . . , Pd+1 ⊂ R be d-dimensional point sets such that the convex hull of each Pi contains the origin. We call the sets Pi color classes, and we think of the points in Pi as having color i. A colorful choice is a set with at most one point from each color class. The colorful Carathéodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far,...

On the Exact Constant i the Quantitative Steinitz Theorem in the Plane

We determine the maximal value of r with the following property. If the convex hull of a set in R 2 contains a unit circle B, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radius r concentric with B. That the result is sharp is shown by the example when the original set is the set of vertices of a regular pentagon circumscribe...