1 research outputs found
On a problem by Dol'nikov
In 2011 at an Oberwolfach workshop in Discrete Geometry, V. Dol'nikov posed
the following problem. Consider three non-empty families of translates of a
convex compact set in the plane. Suppose that every two translates from
different families have a point of intersection. Is it always true that one of
the families can be pierced by a set of three points?
A result by R. N. Karasev from 2000 gives, in fact, an affirmative answer to
the "monochromatic" version of the problem above. That is, if all the three
families in the problem coincide. In the present paper we solve Dol'nikov's
problem positively if is either centrally symmetric or a triangle, and show
that the conclusion can be strengthened if is an euclidean disk. We also
confirm the conjecture if we are given four families satisfying the conditions
above.Comment: 17 pages, 8 figures. Section 4 will be removed from journal versio