The AR(yïl) spaces have been characterized as the r-images of convex sets in normed linear spaces, and the ANR(3K) spaces as the r-images of open subsets of such sets. (The r-images of a space correspond to the retracts of the space. See Borsuk [l ] for proofs of most of the results referred to here.) In this paper we present new characterizations of these spaces. The characterizations yield so...

It is shown that for every k and every p ≥ q ≥ d + 1 there is a c = c(k, p, q, d) < ∞ such that the following holds. For every family H whose members are unions of at most k compact, convex sets in R in which any set of p members of the family contains a subset of cardinality q with a nonempty intersection there is a set of at most c points in R that intersects each member of H. It is also show...

This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in R to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself.

We consider an arrangement A of n hyperplanes in R and the zone Z in A of the boundary of an arbitrary convex set in R in such an arrangement. We show that, whereas the combinatorial complexity of Z is known only to be O ( nd−1 log n ) [3], the outer part of the zone has complexity O ( nd−1 ) (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of...

Separoids capture the combinatorial structure which arises from the separations by hyperplanes of a family of convex sets in some Euclidian space. Furthermore, as we prove in this note, every abstract separoid S can be represented by a family of convex sets in the (|S| − 1)dimensional Euclidian space. The geometric dimension of the separoid is the minimum dimension where it can be represented a...

We introduce regularity classes which are adapted to the most important operations in convexity theory. They are typically between C and C.

For a pair of convex bodies K and K ′ in Ed , the d-dimensional intersections K ∩ (x + K ′), x ∈ Ed , are centrally symmetric if and only if K and K ′ are represented as direct sums K = R ⊕ P and K ′ = R′ ⊕ P ′ such that: (i) R is a compact convex set of some dimension m, 0 ≤ m ≤ d, and R′ = z − R for a suitable vector z ∈ Ed , (ii) P and P ′ are isothetic parallelotopes, both of dimension d − m.

We characterize heirs of so called box types of a polynomially bounded o-minimal structure M . A box type is an n-type of M which is uniquely determined by the projections to the coordinate axes. From this, we deduce various structure theorems for subsets of M, definable in the expansion M of M by all convex subsets of the line. Moreover we obtain a model completeness result for M .

We give a short proof of the theorem that any family of subsets of R with the property that the intersection of any non empty nite subfamily can be represented as the disjoint union of at most k closed convex sets has Helly number at most k d

We introduce the concept of attainable sets of payoffs in two-player repeated games with vector payoffs. A set of payoff vectors is called attainable by a player if there is a finite horizon T such that the player can guarantee that after time T the distance between the set and the cumulative payoff is arbitrarily small, regardless of the strategy Player 2 is using. We provide a necessary and s...