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Register a new userA minimum principle for potentials with application to Chebyshev constants
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For Riesz-like kernels $K(x,y)=f(|x-y|)$ on $Atimes A$, where $A$ is a compact $d$-regular set $Asubset mathbb{R}^p$, we prove a minimum principle for potentials $U_K^mu=int K(x,y)dmu(x)$, where $mu$ is a Borel measure supported on $A$. Setting $P_K(mu)=inf_{yin A}U^mu(y)$, the $K$-polarization of $mu$, the principle is used to show that if ${ u_N}$ is a sequence of measures on $A$ that converges in the weak-star sense to the measure $ u$, then $P_K( u_N)to P_K( u)$ as $Nto infty$. The continuous Chebyshev (polarization) problem concerns maximizing $P_K(mu)$ over all probability measures $mu$ supported on $A$, while the $N$-point discrete Chebyshev problem maximizes $P_K(mu)$ only over normalized counting measures for $N$-point multisets on $A$. We prove for such kernels and sets $A$, that if ${ u_N}$ is a sequence of $N$-point measures solving the discrete problem, then every weak-star limit measure of $ u_N$ as $N to infty$ is a solution to the continuous problem.
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