Geometric estimation of a potential and cone conditions of a body

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.