On the increase of capacity with asymmetry (Q934541)

The paper is concerned with the following question: denote by \[ \alpha(F) := \inf_{y\in\mathbb R^N} \frac{\text{Leb}(F\setminus B_{\rho_F}(y))}{\text{Leb}(F)} \] -- the radius \(\rho_F\) is such that \(\text{Leb}(F) = \text{Leb}(B_1(0)) \rho_F^N\) with \(B_1(0)\subset B_1(0)\) -- a certain modulus of asymmetry measuring the distance of a set \(F\) from a ball. Denote by \(\text{cap}\) the usual capacity in \(\mathbb R^N\), \(N\geq 3\). The main result of the paper is the following lower bound \[ \frac{\text{cap}(E)}{(N-2)\sigma_N} \geq 1 + \frac{4(N-2)}{N^2}\,\left[1-\frac 13\,(7N+2)\varepsilon + \mathcal O(\varepsilon^2)\right]\,\alpha(E)^2 \] for every set \(E\) which is squeezed in between two balls centered at the origin and of radius \(1-\varepsilon_0\) and \(1+\varepsilon_0\) and which is otherwise starlike with respect to every point from some ball of radius \(\varepsilon_1\) centered at \(0\); here \(\varepsilon=\varepsilon_0+\varepsilon_1\). For convex \(E\) a stronger result is known [\textit{R. R. Hall, W. K. Hayman} and \textit{A. W. Weitsman}, J. Anal. Math. 56, 87--123 (1991; Zbl 0747.31004)], but for non-compact sets the present result is near optimal.