First we recall some standard definitions. By a d-dimensional convex body we mean a compact convex subset of Rd with non-empty interior. Two subsets of Rd with non-empty interiors are non-overlapping… (More)

This paper deals with three grading entropy-based rules that describe different soil structure stability phenomena: an internal stability rule, a filtering rule and a segregation rule. These rules… (More)

Let Sd be a d-dimensional simplex in Rd , and let H be an affine hyperplane of Rd . We say that H is a medial hyperplane of Sd if the distance between H and any vertex of Sd is the same constant. The… (More)

In this paper we study how can one generalize the well-known Sylvester theorem for congruent circles. We prove that for any .nite set of at least two points in the plane which has diameter at most √… (More)

The Hadwiger number H(K) of a d-dimensional convex body K is the maximum number of mutually nonoverlapping translates of K that can touch K. We define H∗(K) analogously, with the restriction that all… (More)

We show that the maximum number of mutually nonoverlapping translates of any tetrahedron T which touch T is 18. Moreover, in the case of 18 touching translates the arrangement turns out to be unique.… (More)

The translative kissing number H(K ) of a d-dimensional convex body K is the maximum number of mutually nonoverlapping translates of K which touch K . In this paper we show that there exists an… (More)

For a convex body K, let us denote by t(K) the largest number for which there exists a packing with finitely many translates of K in which every translate has at least t(K) neighbours. In this paper… (More)