# Search Results

## Abstract

K. Bezdek and T. Odor proved the following statement in [1]: If a covering of*E*
^{3} is a lattice packing of the convex compact body*K* with packing lattice Λ (*K* is a Λ-parallelotopes) then there exists such a 2-dimensional sublattice Λ′ of Λ which is covered by the set ∪(*K*+z∣z ∈ Λ′). (*K* ∪*L*(Λ′) is a Λ′-parallelotopes). We prove that the statement is not true in the case of the dimensions*n*=6, 7, 8.

## Abstract

We say that a convex set *K* in ℝ^{
d
}
*strictly separates* the set *A* from the set *B* if *A* ⊂ int(*K*) and *B* ⋂ cl *K* = ø. The well-known Theorem of Kirchberger states the following. If *A* and *B* are finite sets in ℝ^{
d
} with the property that for every *T* ⊂ *A*⋃*B* of cardinality at most *d* + 2, there is a half space strictly separating *T* ⋂ *A* and *T* ⋂ *B*, then there is a half space strictly separating *A* and *B*. In short, we say that the *strict separation number* of the family of half spaces in ℝ^{
d
} is *d* + 2.
In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp.,
similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar
convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for
certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation
number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.

## Abstract

A convex body *K* in ℝ^{
d
} is said to be reduced if the minimum width of each convex body properly contained in *K* is strictly smaller than the minimum width of *K*. We study the question of Lassak on the existence of reduced polytopes of dimension larger than two. We show that a pyramid
of dimension larger than two with equal numbers of facets and vertices is not reduced. This generalizes the main result from
[8].

## Abstract

*N*) and q = (q1, q2, …, q

*N*) are two configurations in

^{ d }(p

_{ i },

*r*

_{ i }) and B

^{ d }(q

_{ i },

*r*

_{ i }) of radius

*r*

_{ i }, for

*i*= 1, …,

*N*. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For

*d*= 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over

## Abstract

Let *K* be a convex body in ℝ^{
d
}, let *j* ∈ {1, …, *d*−1}, and let *K*(*n*) be the convex hull of *n* points chosen randomly, independently and uniformly from *K*. If ∂*K* is *C*
_{+}
^{2}, then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂*K* is *C*
_{+}
^{3}) for the difference of the *j*th intrinsic volume of *K* and the expectation of the *j*th intrinsic volume of *K*(*n*). We extend this formula to the case when the only condition on *K* is that a ball rolls freely inside *K*.

In 1944, Santaló asked about the average number of normals through a point of a given convex body. Since then, numerous results appeared in the literature about this problem. The aim of this paper is to add to this list some new, recent developments. We point out connections of the problem to static equilibria of rigid bodies as well as to geometric partial differential equations of surface evolution.