Anchored hyperplane location problems

The anchored hyperplane location problem is to locate a hyperplane passing through some given points P IR^n and minimizing either the sum of weighted distances (median problem), or the maximum weighted distance (center problem) to some other points Q IR^n . If the distances are measured by a norm, it will be shown that in the median case there exists an optimal hyperplane that passes through at least n - k affinely independent points of Q, if k is the maximum number of affinely independent points of P. In the center case, there exists an optimal hyperplane which isatmaximum distance to at least n - k + 1 affinely independent points of Q. Furthermore, if the norm is a smooth norm, all optimal hyperplanes satisfy these criteria. These new results generalize known results about unrestricted hyperplane location problems