2-dimensional Convexity Numbers and P4-free Graphs

For S ⊆ ℝn a set C ⊆ S is an m-clique if the convex hull of no m-element subset of C is contained in S. We show that there is essentially just one way to construct a closed set S ⊆ ℝ2 without an uncountable 3-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in ℝ2 without uncountable 3-cliques in terms of clopen, P4-free graphs on Polish spaces.

On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.