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

An Explicit Constant for Sullivan's Convex Hull Theorem

An Explicit Counterexample to the Equivariant K = 2 Conjecture

We construct an explicit example of a geometrically finite Kleinian group G with invariant component Ω in the Riemann sphere Ĉ such that any quasiconformal map from Ω to the boundary of the convex hull of Ĉ − Ω in H3 which extends to the identity map on their common boundary in Ĉ, and which is equivariant under the group of Möbius transformations preserving Ω, must have maximal dilatation K > 2...

Semiconvex Hulls of Quasiconformal Sets

We make some remarks concerning the p-semiconvex hulls of the quasiconformal sets, using a recent significant observation of T. Iwaniec in the paper [7] on the important relation between the regularity of quasiregular mappings in the theory of geometric functions and the notion of Morrey’s quasiconvexity in the calculus of variations. We also point out several partial results on a conjecture in...

Lipschitz Spaces and Harmonic Mappings

In [11] the author proved that every quasiconformal harmonic mapping between two Jordan domains with C, 0 < α ≤ 1, boundary is biLipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two Jordan domains Ωj , j = 1, 2, with C, j = 1, 2 boundary is bi-Lipschitz.

Scribe: Thành Nguyen

In this lecture note, we describe some properties of convex sets and their connection with a more general model in topological spaces. In particular, we discuss Tverberg's theorem, Borsuk's conjecture and related problems. First we give some basic properties of convex sets in R d. a i is also in S. Definition 2. Convex hull of a set A, denoted by conv(A), is the set of all convex combination of...