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When C is a ball in ${\Bbb R}^d$ and S is the sphere $\partial C$ , we say that S supports a convex body B if S intersects B and either $B\subseteq C$ (then S is a far support) or the interior of C is disjoint from B (then S is a near support). The focus here is on common supports for a system $\cal B$ of d+1 bodies in ${\Bbb R}^d$ such that for each way of selecting a point from each member of ${\cal B}$ , the selected points are affinely independent and hence form the vertex-set of a d-simplex. The main result asserts that if $({\cal B}',{\cal B}'')$ is an arbitrary partition of ${\cal B}$ , then there exists a unique Euclidean sphere that is simultaneously a near support for each member of ${\cal B}'$ and a far support for each member of ${\cal B}''$ .

We construct a family ofn disjoint convex set in ?d having (n/(d?1))d?1 geometric permutations. As well, we complete the enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane, settle the case for cubes in ?d, and construct a family ofd+1 translates in ?d admitting (d+1)!/2 geometric permutations.

A family of convex sets in the plane admits a common transversal if there is a straight line which intersects (cuts) each member of the family. It is shown that there is a positive integer k such that for any compact convex set C in the plane and for any finite family A \mathcal {A} of pairwise disjoint translates of C: If each 3-membered subfamily of A \mathcal {A} admits a common transversal then there is a subfamily B \mathcal {B} of A \mathcal {A} such that B \mathcal {B} admits a common transversal and | A ∖ B | ⩽ k |\mathcal {A}\backslash \mathcal {B}| \leqslant k .

The object of this paper is to study how many essentially different common transversals a family of convex sets on the plane can have. In particular, we consider the case where the family consists of pairwise disjoint translates of a single convex set.

Let F be a planar family of pairwise disjoint translates of a parallelogram. It is shown that if every four members of F are intersected by some straight line, then there is a straight line which intersects all but at most two members of F. If it is instead assumed that every two members of F are intersected by some line that is parallel to one of the edges of the parallelogram, then the same conclusion holds.

The object of this paper is to study how many essentially different common transversals a family of convex sets on the plane can have. In particular we consider the case where the family consists of pairwise disjoint translates of a single convex set.