Over the last decades considerable progress has been made in establishing reverse (affine) isoperimetric inequalities, i.e., inequalities which usually have simplices or, in the symmetric case, cubes and their polars, as extremals. On the other hand, the notion of Wulff shape has its origins in the classical theory of crystal growth, and provides a unifying setting for several extremal problems with an underlying isotropic measure in more modern mathematical terms. Based on these, some sharp reverse volume inequalities for origin-symmetric Wulff shapes and their polars were obtained by Lutwak, Yang and Zhang, which generalize several of the previously obtained Ball-Barthe volume ratio inequalities for unit balls of subspaces of \(L_p\). But the problem of finding similar volume estimates for not necessarily origin-symmetric Wulff shapes remained open.
In this paper, the authors establish such sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars including a complete description of all equality cases, along with the characterization of all extremals. Firstly, they establish a sharp bound for the volume of the Wulff shape determined by an f-centered isotropic measure, which yields an explicit description of how the displacement enters the sharp upper bound for the volume of Wulff shape. As a natural dual to the above result, the authors then provide a sharp lower bound for the volume of the polar of the Wulff shape, which is independent of the displacement of the Wulff shape. These two results are based on a refinement of the approach towards recently established simplex inequalities by Lutwak, Yang and Zhang, which, in turn, uses many ideas of Ball and Barthe, and can also be obtained by applications of Barthe’s continuous Brascamp-Lieb inequality and its inverse. Finally, the authors show that these new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang and Zhang, which provide a partial solution to a problem by Zhang.