On the $p$-harmonic radii of circular sectors

It is proved that the property of logarithmic concavity of the conformal radius of a circular sector (considered as a function of the angle) extends to the domains of Euclidean space. In this case, the conformal radius is replaced by $p$-harmonic one, and the fundamental solution of the Laplace $p$-equation acts as logarithm. In the case of $p=2$, the presence of an asymptotic formula for the capacity of a degenerate condenser allows us to generalize this result to the case of a finite set of points. The method of the proof leads to the solution of one particular case of an open problem of A. Yu. Solynin.