A stability result for mean width of Lp-centroid bodies

We give a different proof of a recent result of Klartag [B. Klartag, A central limit theorem for convex sets, Invent. Math. 168 (1) (2007) 91–131] concerning the concentration of the volume of a convex body within a thin Euclidean shell and proving a conjecture of Anttila, Ball and Perissinaki [M. Anttila, K. Ball, I. Perissinaki, The central limit problem for convex bodies, Trans. Amer. Math. Soc. 355 (12) (2003) 4723–4735]. It is based on the study of the Lp-centroid bodies. We prove an almost isometric reverse Hölder inequality for their mean width and a refined form of a stability result.