  The concept of $H$-sets as introduced by Collatz in 1956 was very useful in
univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the
multivariate setting, the situation is much worse, because there is no
alternation, and $H$-sets exist, but are only rarely accessible by mathematical
arguments. However, in Reproducing Kernel Hilbert spaces, $H$-sets are shown
here to have a rather simple and complete characterization. As a byproduct, the
strong connection of $H$-sets to Linear Programming is studied. But on the
downside, it is explained why $H$-sets have a very limited range of
applicability in the times of large-scale computing.
