Bounded linear regularity, the strong conical hull intersection property (strong CHIP), and the conical hull intersection property (CHIP) are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. It was shown recently that these properties are fundamental in several branches of convex optimization, including convex feasibility problems, error bounds, Fe...

Let K ⊂ R be a smooth convex set and let Pλ be a Poisson point process on R of intensity λ. The convex hull of Pλ ∩ K is a random convex polytope Kλ. As λ → ∞, we show that the variance of the number of k-dimensional faces of Kλ, when properly scaled, converges to a scalar multiple of the affine surface area of K. Similar asymptotics hold for the variance of the number of k-dimensional faces fo...

Let V be a real linear space. The functor ConvexComb(V ) yielding a set is defined by: (Def. 1) For every set L holds L ∈ ConvexComb(V ) iff L is a convex combination of V . Let V be a real linear space and let M be a non empty subset of V . The functor ConvexComb(M) yielding a set is defined as follows: (Def. 2) For every set L holds L ∈ ConvexComb(M) iff L is a convex combination of M . We no...

The (n, k)-hypersimplex is the convex hull of all 0/1-vectors of length n with coordinate sum k. We explicitly determine the extension complexity of all hypersimplices as well as of certain classes of combinatorial hypersimplices. To that end, we investigate the projective realization spaces of hypersimplices and their (refined) rectangle covering numbers. Our proofs combine ideas from geometry...

We study the facial structure and Carathéodory number of the convex hull of an orbit of the group of rotations in R acting on the space of pairs of anisotropic symmetric 3× 3 tensors. This is motivated by the problem of determining the structure of some proteins in aqueous solution.

We elaborate on the use of shadow systems to prove a particular case of the conjectured lower bound of the volume product P(K) = minz∈int(K) |K|||K|, where K ⊂ R is a convex body and K = {y ∈ R : (y − z) · (x − z) 6 1 for all x ∈ K} is the polar body of K with respect to the center of polarity z. In particular, we show that if K ⊂ R is the convex hull of two 2-dimensional convex bodies, then P(...

Let P be a set of n points on the plane in general position. We say that a set Γ of convex polygons with vertices in P is a convex decomposition of P if: Union of all elements in Γ is the convex hull of P, every element in Γ is empty, and for any two different elements of Γ their interiors are disjoint. A minimal convex decomposition of P is a convex decomposition Γ′ such that for any two adjac...

Let m > 1 be an integer, Bm the set of all unit vectors of R pointing in the direction of a nonzero integer vector of the cube [−1, 1]. Denote by sm the radius of the largest ball contained in the convex hull of Bm. We determine the exact value of sm and obtain the asymptotic equality sm ∼ 2 √ log m . Primary subject classification: 52B11 Secondary subject classification: 52B12 §

Convexity predicates and the convex hull operator continue to play an important role in theories of spatial representation and reasoning, yet their firstorder axiomatization is still a matter of controversy. In this paper, we present a new approach to adding convexity to a mereotopological theory with boundary elements by specifying first-order axioms for a binary segment operator s. We show th...

We show that half-integral polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω(log n/ log log n) with positive probability — even if the size of the set relative to the total number of half-integral points of the cube tends to 0. The high rank is due to certain obstructions. We determine the exact threshold number, when these obst...