Potential Theory Of The Farthest-Point Distance Function
| Citations: | 5 - 4 self |
BibTeX
@MISC{Laugesen_potentialtheory,
author = {Richard S. Laugesen and Igor and Igor E. Pritsker},
title = {Potential Theory Of The Farthest-Point Distance Function},
year = {}
}
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Abstract
We study the farthest{point distance function, which measures the distance from z 2 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 E has many interesting properties that reect the topology and geometry of the compact set E. We prove E (E) 2 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 E (E) = 2 for smooth convex sets of constant width. We conjecture E (E) 2 for all E. 1.
Citations
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| 96 | Classical potential Theory and its Probabilistic Counterpart - Doob - 1984 |
| 96 | Potential theory in the complex plane - Ransford - 1995 |
| 77 | and complex analysis. 3rd edition - Rudin, Real - 1986 |
| 11 | Convex bodies of constant width - Chakerian, Groemer - 1983 |
| 10 | Sharp inequalities for the product of polynomials - Boyd - 1994 |
| 10 | Distance to C k hypersurfaces - Krantz, Parks |
| 8 | Fractal geometry - Krantz - 1989 |
| 8 | Products of polynomials in uniform norms - Pritsker |
| 6 | Convex bodies of constant width, in Convexity and its applications - Chakerian, Groemer - 1983 |
| 1 | E-mail address : laugesen@math.uiuc.edu - Press - 1994 |
