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.

Appendix: A Lower Bound for \(\tilde{R}(-1, \pi ,+\infty , {{\mathrm{diam}}}\Omega ,R_0)\)

Let $\Omega$ be a convex body in $\mathbb {R}^m$. Thanks to the convexity of $\Omega$, we can take the uniform boundary inner cone of the complement of $\Omega$ as a half-space. Let $0< R_0 < {{\mathrm{diam}}}\Omega$. In this appendix, we give a lower bound for $\tilde{R} = \tilde{R}(-1, \pi , +\infty ,{{\mathrm{diam}}}\Omega ,R_0)$ given in Lemma 3.8 with $\alpha =-1, \kappa = \pi , \delta =+\infty$, and $D={{\mathrm{diam}}}\Omega$. Let us estimate the root of the function

$$\begin{aligned} E (R) = \left( \int _{C\left( Re_1; \pi ,+\infty \right) \cap B_{{{\mathrm{diam}}}\Omega }(0)} -\int _{B_{{{\mathrm{diam}}}\Omega }(0) \setminus B_{R_0}(0)} \right) \left| \right. \xi \left| \right. ^{-(m+1)} d \xi . \end{aligned}$$

(6.1)

Here, we remark that, from Lemma 3.8 (3), if $R\ge R_0$, then $E(R)<0$. Thus we may assume $R < R_0$ in what follows.

Let $\varphi (R)=\arccos (R/{{\mathrm{diam}}}\Omega )$. Using polar coordinates, we obtain

$$\begin{aligned} E(R)&=\sigma _{m-2} \left( S^{m-2}\right) \int _0^{\varphi (R)} \left( \int _{R/\cos \theta }^{{{\mathrm{diam}}}\Omega } r^{-2}dr \right) \sin ^{m-2}\theta d\theta \nonumber \\&\quad -\sigma _{m-2} \left( S^{m-2}\right) \int _0^\pi \left( \int _{R_0}^{{{\mathrm{diam}}}\Omega } r^{-2} dr\right) \sin ^{m-2}\theta d\theta \nonumber \\&=\frac{\sigma _{m-2} \left( S^{m-2}\right) }{R} \left( \frac{\sin ^{m-1} \varphi (R)}{m-1} +\frac{R}{{{\mathrm{diam}}}\Omega } \int _{\varphi (R)}^\pi \sin ^{m-2}\theta d\theta \right. \nonumber \\&\quad \left. -\frac{R}{R_0}\int _0^\pi \sin ^{m-2}\theta d\theta \right) \nonumber \\&=: \frac{\sigma _{m-2} \left( S^{m-2}\right) }{R}f(R). \end{aligned}$$

(6.2)

Direct computation shows the following properties:

$$\begin{aligned} f'(R)&=\frac{1}{{{\mathrm{diam}}}\Omega } \int _{\varphi (R)}^\pi \sin ^{m-2}\theta d\theta -\frac{1}{R_0} \int _0^{\pi } \sin ^{m-2}\theta d\theta <0,\end{aligned}$$

(6.3)

$$\begin{aligned} f''(R)&=-\frac{\varphi '(R) \sin ^{m-2} \varphi (R)}{{{\mathrm{diam}}}\Omega } >0. \end{aligned}$$

(6.4)

Since we have

$$\begin{aligned} f'(0)= \left( \frac{1}{{{\mathrm{diam}}}\Omega } -\frac{2}{R_0} \right) \int _0^{\pi /2} \sin ^{m-2}\theta d\theta , \end{aligned}$$

(6.5)

we obtain

$$\begin{aligned} \tilde{R} \left( -1, \pi ,+\infty ,{{\mathrm{diam}}}\Omega , R_0\right)&> -\frac{f(0)}{f'(0)} \nonumber \\&=\frac{1}{\displaystyle (m-1)\left( \frac{2}{R_0}-\frac{1}{{{\mathrm{diam}}}\Omega } \right) \int _0^{\pi /2} \sin ^{m-2}\theta d\theta }\nonumber \\&\ge \frac{R_0}{\displaystyle 2(m-1)\int _0^{\pi /2} \sin ^{m-2}\theta d\theta }. \end{aligned}$$

(6.6)

For example, in the case of $m=2$, the above lower bound coincides with $R_0 / \pi \approx 0.3183 R_0$.