Computational Methods and Function Theory 8 (2008), No. 1, 203--224
Copyright Heldermann Verlag 2008
L. E. Fraenkel
lef@bath.ac.uk , University of Bath, Department of Mathematical Sciences, Claverton Down, Bath BA2 7AY, U.K.
Let $E$ be a compact, starlike set in $\mathbb{R}^N$, $N\geq2$, that is very close to a ball $B$ of the same area or volume. This paper presents inequalities, for logarithmic capacity if $N=2$ or for capacity if $N\geq 3$, of the form $$ \lcap E\geq\exp\{K_2\alpha(E)^2\}\,\lcap\overline{B} \quad\mbox{or}\quad \capac E\geq\{1+K_N\alpha(E)^2\}\,\capac\overline{B}, $$ where $\alpha(E)$ is a modulus of asymmetry that measures the departure of the shape of $E$ from that of $B$. The results are far less general than those of Hansen and Nadirashvili for $N=2$ and those of Hall, Hayman and Weitsman for $N\geq3$, but (for the particular sets considered) the present inequalities are somewhat sharper.
Keywords: Capacity, isoperimetric inequalities, potential theory.
MSC 2000: 31C15, 35J85.
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