A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a “fat” one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm’s theorem and its proof on illuminating con...

We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: “given an x ∈ V , does x belong to X?” can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, ...

Given a convex body K , consider the smallest number N so that there is a point P ∈ ∂K such that every circle centred at P intersects ∂K in at most N points. In 1946 Erdős conjectured that N = 2 for all K , but there are convex bodies for which this is not the case. As far as we know there is no known global upper bound. We show that no convex body has N = ∞ and that there are convex bodies for...

In this paper we study Euclidean projections of a p-convex body in IR. Precisely, we prove that for any integer k satisfying lnn ≤ k ≤ n/2, there exists a projection of rank k with the distance to the Euclidean ball not exceeding Cp(k/ln(1 + nk )) 1/p−1/2, where Cp is an absolute positive constant depending only on p. Moreover, we obtain precise estimates of entropy numbers of identity operator...

Coarse-grained modeling of molecular fluids is often based on nonspherical convex rigid bodies like ellipsoids or spherocylinders representing rodlike or platelike molecules or groups of atoms, with site-site interaction potentials depending both on the distance among the particles and the relative orientation. In this category of potentials, the Gay-Berne family has been studied most extensive...

Recall that an element de of the standard measure is written as de = dp · dξ, where (p, ξ) is the usual parametrization of a plane e: p is the distance of e from the origin O; ξ ∈ S is the direction normal to e, dξ is an element of solid angle of the unit sphere S. Where appropriate we write h(e) = h(p, ξ). The concept of a flag in R which naturally emerges in Combinatorial integral geometry wi...

In this paper a Blaschke-Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of set

Supported by NSF grant ECS-8418392 I n t r o d u c t i o n The Integer Programming (feasibility) Problem is the problem of determining whether there is a vector of integers satisfying a given system of linear inequalities. In settling an important open problem, H.W.Lenstra (1981,1983) showed in an elegant way that when n the number of variables is fixed, there is a polynomial t ime algorithm to...