convex sets

Javier Alonso*, Pedro Mart́ın. Universidad de Extremadura, Badajoz, Spain. Characterizations of ellipsoids by sections. Let S be the boundary of a convex body in the d-dimensional Euclidean space E (d ≥ 3). It is well known that S is an ellipsoid if and only if the section of S given by any hyperplane is ellipsoidal. The question of whether it is actually necessary to consider “any” hyperplane to characterize S as ellipsoid or is enough to consider “some” hyperplanes is at the origin of an important family of characterizations of ellipsoids. In that context, we study whether we can restrict the hyperplanes to those that are parallel to two or three fixed hyperplanes and also whether we can consider only hyperplanes that contain one of two fixed linear varieties. Imre Bárány*, Alfredo Hubard, Jerónimo Jesus. Alfréd Rényi Institute of Mathematics, Budapest, Hungary. Slicing convex sets and measures by a hyperplane. Convex bodies K1, . . . , Kd ⊂ R are said to be well separated if aff{x1, . . . , xd} is a nondegenerate hyperplane for every x1 ∈ K1, . . . , xd ∈ Kd. The main result in this talk says that if K1, . . . , Kd are well separated convex bodies in R and α1, . . . , αd ∈ [0, 1], then there exists a unique oriented halfspace, H, such that |H ∩ Ki| = αi|Ki| for every i = 1, . . . , d, where |K| denotes the volume of the convex body K. The result is extended from convex bodies to measures. András Bezdek. Auburn University, Auburn, AL. On iterative processes generating dense point sets. There are several results in the literature concerning iterative processes in the plane. A typical problem starts with the description of a geometric construction, which when applied to an initial point set generates larger point sets. The problem usually is to prove that repeated expansions lead to an everywhere dense point set. We refer to D. Ismailescu, who started to investigate the construction “add the circumcenters (CC) (incenters (IC), orthocenters (OC) respectively) of all nondegenerate triangles formed by existing points.” In a joint paper Iorio, Ismailecsu, Radoicic and Silva solved the planar IC and the planar CC problem and stated conjectures concerning the planar OC problem. The talk outlines the solutions of the following versions: planar OC problem (with G. Ambrus, 2005) 3-dimensional IC problem (with G. Ambrus, 2006) hyperbolic and spherical IC problem (with T. Bisztriczky, 2006) iterative processes in lattices (2007). Ted Bisztriczky. University of Calgary, Calgary, Canada. Classification of bicyclic 4-polytopes.