Two of the most important affine isoperimetric inequalities are the Blaschke-Santalo inequality (see, e.g., Gardner [6, p. 322] or Schneider [12, p. 425]) and the classical affine isoperimetric inequality of affine differential geometry (see, e.g., Schneider [12, p. 419]). These two inequalities are closely related in that given either one of these inequalities, then by well-known methods one can be quickly deduced from the other. The aim of this article is to establish a new family of analytic inequalities and their geometric counterparts. One of the members of this family of inequalities turns out to be the Blaschke-Santalo inequality.
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Lutwak et al. (1997) studied this question.