A new scale-invariant geometry on L1 spaces Original Research Article

Nieto, Torres and Vázquez-Trasande recently considered, for any nn∈NN, the bivariate map View the MathML sourced:[0,1]n×[0,1]n→R:(p,q)↦||p−q||1||max⁡(|p|,|q|)||1 Turn MathJax on and showed that this map satisfies the triangle inequality. Here, we suggest to consider, more generally, for any space X with a positive measure μ, the bivariate map View the MathML sourcedNTV:L1(X,µ)×L1(X,µ)→R:(p,q)↦∫x|p−q|dµ∫xmax⁡(|p|,|q|)dµ Turn MathJax on and show that, for nonnegative maps p, q L1 (X, μ), the length of a geodesic (relative to d) coincides with the sum View the MathML sourceIn∫Xmax⁡(p,q)dµ∫Xpdµ+In∫Xmax⁡(p,q)dµ∫Xqdµ. Turn MathJax on In particular, a continuous path S : Z0, 1] → L1 (X, μ)+: t St is a geodesic in L1 (X, μ) relative to this metric if and only if there exists some t0 Z0, 1] with S{int0} = max(p, q), and one has St ≤ St′ for all t, t′ Z0, t0] with t ≤ t′ and for all t, t′ Zt0, 1] with t ≥ t′.