The "farthest point distance
function" of a non-empty compact set E in the plane is defined by
d_{E}(x) := max_{y\in E} |x-y|. This function arises naturally in the
study of sharp inequalities for norms of products of polynomials. The
crucial observation which makes it useful is that log d_{E}(x) is a
logarithmic potential: it can be written as $\int \log |x-y| d\sigma
_{E}(y)$ for some probability measure $\sigma_{E}.$ However, the
distribution of the measure $\sigma_{E}$ has proved mysterious. On the
basis of various examples Laugesen and Pritsker were led to the
unexpected and intriguing conjecture that $\sigma_{E}(E)\leq 1/2$
whenever E contains more than one point. This talk will use a blend of
potential theory and convex geometry to explain why this inequality is
true, and when equality holds. This is joint work with Ivan Netuka
(Charles University, Prague).