A Min-Max Theorem for Sums of Translates of a Function

Given a partition R of w0, 1x: 0sr1Fr2F ??? FrN s1, let f r.sJ r.q nNs1K ryrn., rgw0, 1x, where J r. is concave and K:wy1, 1x_ 04ªR is a concave cusp with its point at 0. Necessary conditions on the partition R are determined when R minimizes max0FrF1f r.smax1FnF Ny1 fn, where fns maxrnFrFrnq1 f r.. It is shown that the partition that minimizes max1FnF Ny1 fn also maximizes min1FnF Ny1 fn.