论文标题
针对某些类别的尖锐构型的最小最大极化
Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere
论文作者
论文摘要
我们考虑在球体上找到$ n $ - 点配置的问题，$ s^d \ subset \ rr^{d+1} $，其绝对最大值的最大值超过其总潜力的$ s^d $。给定配置中的每个点$ {\ bf y} $在s^d $ in s^d $ in s^d $ ins $ f \（\ left | {\ bf x} - {\ bf y} \ y} \ right |^2 \）$（$ [0,0,4]的$ [0,4]中， $ \左| {\ bf x} - {\ bf y} \右| $是点之间的欧几里得距离〜$ {\ bf x} $和$ {\ bf y} $。任何此类配置$ \olΩ_n$的潜力的绝对最大值超过$ s^d $，以$ \olΩ_n$的点获得。
We consider the problem of finding an $N$-point configuration on the sphere $S^d\subset \RR^{d+1}$ with the smallest absolute maximum value over $S^d$ of its total potential. The potential induced by each point ${\bf y}$ in a given configuration at a point ${\bf x}\in S^d$ is $f\(\left|{\bf x}-{\bf y}\right|^2\)$, where $f$ is continuous on $[0,4]$ and completely monotone on $(0,4]$, and $\left|{\bf x}-{\bf y}\right|$ is the Euclidean distance between points~${\bf x}$ and ${\bf y}$. We show that any sharp point configuration $\OLω_N$ on $S^d$, which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over $S^d$ of the potential of any such configuration $\OLω_N$ is attained at points of $\OLω_N$.
