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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.

Blaschke–Santaló Inequalities for Minkowski and Asplund Endomorphisms