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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 $S^d$ ofits total potential. The potential induced by each point ${\bf y}$ in a givenconfiguration at a point ${\bf x}\in S^d$ is $f$\left|{\bf x}-{\bfy}\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 betweenpoints~${\bf x}$ and ${\bf y}$. We show that any sharp point configuration$\OL\omega_N$ on $S^d$, which is antipodal or is a spherical design of an evenstrength is a solution to this problem. We also prove that the absolute maximumover $S^d$ of the potential of any such configuration $\OL\omega_N$ is attainedat points of $\OL\omega_N$.

Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere