Blaschke-Santaló inequalities for Minkowski and Asplund endomorphisms

Abstract

It is shown that each monotone Minkowski endomorphism of convex bodies gives rise to an isoperimetric inequality which directly implies the classical Urysohn inequality. Among this large family of new inequalities, the only affine invariant one -the Blaschke-Santaló inequality -turns out to be the strongest one. A further extension of these inequalities to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover, for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequalities for log-concave functions is established which generalizes the functional Blaschke-Santaló inequality.

A Characterization of the Polarity Mapping for Convex Bodies

“…For more results on the characterization of duality and lattice endomorphism of the class of convex bodies and of convex sets, we can refer to Refs. [25][26][27][28][29].…”

Fixed points of Minkowski valuations

“…Over the past 15 years, it has become more and more apparent that several classic inequalities involving projection bodies (of arbitrary degree) hold, in fact, for the entire class, or at least a large subclass, of Minkowski valuations intertwining rigid motions (see, e.g., [5,7,20,23,43,53]). Among the first results in this direction, it was proved in [53] that if Φ 1 : K n → K n is a non-trivial continuous translation invariant Minkowski valuation of degree 1 which commutes with SO(n) and is monotone w.r.t.…”

Iterations of Minkowski Valuations

“…In recent years, several classic inequalities involving projection bodies of arbitrary degree have been shown to hold for large (if not all) subclasses of Minkowski valuations intertwining rigid motions (see, e.g., [5,6,19,21,36,43]). Some of these results are indeed a consequence of already known inequalities for the projection bodies, which turn out to be the limiting cases of such families of inequalities.…”