Geometric Estimation of a Potential and Cone Conditions of a Body

Abstract

We investigate a potential obtained as the convolution of a radially symmetric function and the characteristic function of a body (the closure of a bonded open set) with exterior cones. In order to restrict the location of a maximizer of the potential into a smaller closed region contained in the interior of the body, we give an estimate of the potential using the exterior cones of the body. Moreover, we apply the result to the Poisson integral for the upper half space.

Stationary radial centers and symmetry of convex polytopes

“…In [21], it was shown that if h ≥ √ m + 2 diam Ω, then Ω has a unique illuminating center, and that, as h goes to infinity, the unique illuminating center converges to the centroid of Ω. In [23], it was shown that the limit point of any convergent sequence of illuminating centers c(h j ) of height h j with h j → 0 + is an r −1−m -center. Thus an illuminating center moves with respect to h in general.…”