The cross covariogram gK,L of two convex sets K and L in R n is the function which associates to each x ∈ R the volume of K ∩ (L + x). Very recently Averkov and Bianchi [AB] have confirmed Matheron’s conjecture on the covariogram problem, that asserts that any planar convex body K is determined by the knowledge of gK,K . The problem of determining the sets from their covariogram is relevant in ...

Let L be a Lie group and A a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L and G---A = L. Let a~G be such that Ada is semisimple and not contained in a compact subgroup of Aut (Lie(G)). Consider the expanding horospherical subgroup of G associated to a defined as U + = {g~ G:a-"ga"~ e as n ~ oo}. Let f2 be a non-empty open subset of U § and nf-* ~ be ...

Let v be a (standardized) absolute norm on en. A matrix H in enn is called normHermitian jf the numerical range V(H) determined by v is real. Let :re be the set of all norm-Hermitians in en"' We determine an equivalence relation'" on {t, .•. , n} with the following property: Let HE en"' Then HE :re if and only if H is Hermitian and h,) = 0 if i + j. Let,l =.# + i:lC. Then.l is a subalgebra of e...

1. The concept of arcwise convex set is a natural generalization of the notion of L set. The latter was studied by Horn and Valentine [2]1. It is especially interesting to observe how the theorem about the complement of an arcwise convex continuum sheds light on the corresponding theorem for L sets. In order to make this precise, the following definitions are used. Definition 1. An arc C is sai...

We give a description of an affine mapping T involving contact pairs of two general convex bodies K and L, when T (K) is in a position of maximal volume in L. This extends the classical John’s theorem of 1948, and is applied to the solution of a problem of Grünbaum; namely, any two convex bodies K and L in Rn have non-degenerate affine images K ′ and L′ such that K ′ ⊂ L′ ⊂ −nK ′. As a corollar...

Let \(1 \to K G\,\mathop \limits^\pi \,Q\) be an exact sequence of hyperbolic groups. Q1 < Q a quasiconvex subgroup and let G1 = ??1(Q1). Under relatively mild conditions (e.g. if is closed surface group or free convex cocompact), we show that infinite index subgroups are in G. Related results proven for metric bundles, developable complexes groups graphs

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder condition for a finitely generated subgroup of A(Γ) that guarantees it is free, undistorted, a...

Properties of several sorts of lattices of convex subsets of Rn are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of Rn and the lattice of all convex subsets of R. The lattices of arbitrary, of open bounded, and of compact convex sets in Rn all satisfy the same identitie...

We define a class of L-convex-concave subsets of RP , where L is a projective line in RP . These are sets whose sections by any plane containing L are convex and concavely depend on this plane. We prove a version of Arnold’s conjecture for these sets, namely we prove that each such set contains a line.