5 research outputs found
On a colorful problem by Dol'nikov concerning translates of convex bodies
In this note we study a conjecture by Jer\'onimo-Castro, Magazinov and
Sober\'on which generalized a question posed by Dol'nikov. Let
be families of translates of a convex compact set in
the plane so that each two sets from distinct families intersect. We show that,
for some , can be pierced by at most points. To
do so, we use previous ideas from Gomez-Navarro and Rold\'an-Pensado together
with an approximation result closely tied to the Banach-Mazur distance to the
square
Packing and covering with balls on Busemann surfaces
In this note we prove that for any compact subset of a Busemann surface
(in particular, for any simple polygon with geodesic metric)
and any positive number , the minimum number of closed balls of radius
with centers at and covering the set is at most 19
times the maximum number of disjoint closed balls of radius centered
at points of : , where and
are the covering and the packing numbers of by -balls.Comment: 27 page