PUB - Publikationen an der Universität Bielefeld

Nieto, Torres and Vazquez-Trasande recently considered, for any n is an element of N, the bivariate map and showed that this map satisfies the triangle inequality. Here, we suggest to consider, more generally, for any space X with a positive measure mu, the bivariate map and show that, for nonnegative maps p, q is an element of L, (X, p), the length of a geodesic (relative to d) coincides with the sum In particular, a continuous path S : [0, 1] --> L-1 (X, mu)(+) : t --> S-t is a geodesic in L-1 (X, mu) relative to this metric if and only if there exists some t(0) is an element of [0, 1] with St(0) = max(p, q), and one has S-t < S-t', for all t, t' is an element of [0, t(o)] with t less than or equal to t' and for all t, t' is an element of [t(0), 1] with t greater than or equal to t'. (C) 2004 Elsevier Ltd. All rights reserved.