Boundaries of compact convex sets and fragmentability

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

A note on convex renorming and fragmentability

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly ...

Algebraic boundaries of convex semi-algebraic sets

We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a com...

Directed Sets and Diierences of Convex Compact Sets 3

Isomorphisms of Spaces of Continuous Affine Functions on Compact Convex Sets with Lindelöf Boundaries

Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and both extX and extY are Lindelöf spaces. We prove that, if there exists an isomorphism T : Ac(X) → Ac(Y ) with ‖T‖ · ‖T‖ < 2, then extX is homeomorphic to extY . This generalizes results of H.B. Cohen and C.H. Chu.