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).