Weights of boundaries of compact convex sets

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...

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 $k$ ...

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.

Directed Sets and Diierences of Convex Compact Sets 3

Spline Subdivision Schemes for Convex Compact Sets

The application of spline subdivision schemes to data consisting of convex compact sets with addition replaced by Minkowski sums of sets is investigated These methods generate in the limit set valued functions which can be expressed explicitly in terms of linear com binations of integer shifts of B splines with the initial data as coe cients The subdivision techniques are used to conclude that ...