We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an N-point configuration that maximizes the minimum value of its potential over a set A in p-dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set A. We find that for Riesz kernels 1/|x − y|^s with s > p − 2 the optimum unconstrained configurations concentrate close to the set A and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained problem on a class of d-rectifiable sets. We also investigate the new unconstrained problem in special cases such as for spheres and balls. In the last section we formulate some natural open problems and conjectures.

我们引入并研究了无约束极化（或切比雪夫）问题，该问题要求找到一个N点配置，该配置在p维欧几里得空间中的集合A上最大化其势的最小值。将该问题与要求点属于集合A的约束问题进行比较。我们发现对于Riesz内核1 / | x − y | ^小号与小号> p - 2最佳不受约束配置集中接近设定甲并且基于这个基本事实，我们恢复了与一类d可整流集上更经典的约束问题相同的极化渐近值。在特殊情况下，例如球体和球，我们还将研究新的无约束问题。在上一节中，我们提出了一些自然的公开问题和猜想。