The effect of dimension on certain geometric problems of irregularities of distribution (Q1338915)

The author addresses the following question as well as similar problems of discrepancy: Let \(p_1, p_2, \dots, p_N\) span the \(t\)- dimensional Euclidean space \(E^t\) and be two-coloured (``red'' and ``blue''). For a hyperplane \(h\) containing none of these points and for an open half-space \(H\) determined by \(h\), let \(D(h)\) denote \(|r(h)- b(h)|\), where \(r(h)\) and \(b(h)\) stand for the numbers of red and blue points lying in \(H\). As a typical result of this paper, the author gives a lower estimate of \(\sup D(h)\), where \(h\) ranges over all hyperplanes. Special emphasis is put upon the dimension \(t\) of the space, which is taken as a variable. In fact, the author considers these problems in a much more general setting. Let \(\nu\) be a signed Borel measure of total mass 0 supported by acompact subset of \(E^t\). Then the separation discrepancy \(D_s (\nu)\) is defined by \[ D_s (\nu):= \sup\{ |\nu (H)|:\;H\text{ an open half-space of } E^t\}. \] Coloured points as above correspond to certain atomic measures. The quantity \(D_s (\nu)\) as well as \(L^2\) averages of \(|\nu (H)|\) are studied, then \(t\) taken as a variable. Interesting results as the above estimate for \(\sup D(h)\) follow as immediate corollaries from the general inequalities.