Convexity on unions of sets
Convexity ranks in higher dimensions
A subset of a vector space is called countably convex if it is a countable union of convex sets. Classification of countably convex subsets of topological vector spaces is addressed in this paper. An ordinal-valued rank function ϱ is introduced to measure the complexity of local nonconvexity points in subsets of topological vector spaces. Then ϱ is used to give a necessary and sufficient condition for countable convexity of closed sets. Theorem. Suppose that S is a closed subset of a Polish linear...
Convexity structures in zero-dimensional compact spaces.
Convexity theories 0 fin. Foundations.
In this paper we study big convexity theories, that is convexity theories that are not necessarily bounded. As in the bounded case (see [4]) such a convexity theory Γ gives rise to the category ΓC of (left) Γ-convex modules. This is an equationally presentable category, and we prove that it is indeed an algebraic category over Set. We also introduce the category ΓAlg of Γ-convex algebras and show that the category Frm of frames is isomorphic to the category of associative, commutative, idempotent...
Convexly independent subsets of the Minkowski sum of planar point sets.
Cooperative guards in art galleries with one hole.
Corner cuts and their polytopes.
Correct Soltan-Vasiloi criterion for convex cones.
Correction to the paper: Packing of 180 equal circles on a sphere. Elemente der Mathematik. Vol. 38, 1983, 119-122.
Correction to the paper: Sets invariant under projections onto one dimensional subspaces
Corrections to my paper "On the geometry of convex reflectors" (Banach Center Publ. 57 (2002), 155-169)
Corrigendum to "Preassigning the Shape for Bodies of Constant Width".
Countable Decompositions of R2 and R3 .
Countably convex sets
We investigate countably convex subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition...
Counting -polytopes with vertices.
Counting Facets and Incidences.
Counting simplexes in .
Counting unrooted maps using tree-decomposition.
Covering a disk by disks.
Covering a plane convex body with five smaller homothetical copies