Although the main result (Theorem 2.1) in [1] is correct, the proof there has a fatal flaw. A correct proof was given by Ni and Zhu [2] (see also [3]).
References
[1] W. Chen, R. Howard, E. Lutwak, D. Yang, and G. Zhang, A generalized affine isoperimetric inequality, J. Geom. Anal. 14 (2004), no. 4, 597--612.
[2] Y. Ni and M. Zhu, One-dimensional conformal metric flow, Adv. Math. 218 (2008), no. 4, 983--1011.
[3] Y. Ni and M. Zhu, Steady states for one-dimensional curvature flows, Commun. Contemp. Math. 10 (2008), no. 2, 155--179.