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Potential Theory of the Farthest-Point Distance Function |
Canad. Math. Bull. 46(2003), 373-387
Published:2003-09-01
Printed: Sep 2003
Printed: Sep 2003
- Richard S. Laugesen
- Igor E. Pritsker
Abstract
We study the farthest-point distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$-gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
| Keywords: | distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width |
| MSC Classifications: |
31A05 - Harmonic, subharmonic, superharmonic functions 52A10 - Convex sets in $2$ dimensions (including convex curves) [See also 53A04] 52A40 - Inequalities and extremum problems |