Geodesic
On the geometry of the unit spheres of the Lorentz spaces Lw,1
Glasgow mathematical journal, Tome 34 (1992) no. 1, pp. 21-25

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We identify the extreme points of the unit sphere of the Lorentz space Lw,1 This yields a characterization of the surjective isometries of Lw,1(0,1). Our main result is that every element in the unit sphere of Lw,1 is the barycenter of a unique Borel probability measure supported on the extreme points of the unit sphere of Lw,1. 
Carothers, N. L.; Dilworth, S. J.; Trautman, D. A. On the geometry of the unit spheres of the Lorentz spaces Lw,1. Glasgow mathematical journal, Tome 34 (1992) no. 1, pp. 21-25. doi: 10.1017/S0017089500008508
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