Affine vs. Euclidean isoperimetric inequalities

Abstract

It is shown that every even, zonal measure on the Euclidean unit sphere gives rise to an isoperimetric inequality for sets of finite perimeter which directly implies the classical Euclidean isoperimetric inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them -the Petty projection inequality. As an application, a family of sharp Sobolev inequalities for functions of bounded variation is obtained, each of which is stronger than the classical Sobolev inequality. Moreover, corresponding families of L p isoperimetric and Sobolev type inequalities are also established.

Fixed points of Minkowski valuations

“…Here, V i , 0 ≤ i ≤ n, denotes the i-th intrinsic volume (see Section 2 for definition) and K|u ⊥ is the orthogonal projection onto the hyperplane u ⊥ . The polar projection inequality of Petty [44], providing the classical relation between the volume of a convex body and its polar projection body, is an affine invariant inequality that not only significantly improves the classical isoperimetric inequality but had a tremendous impact in geometric analysis that can still be felt to this day (see, e.g., [9,19,20,39,40] and the references therein). In view of the Blaschke-Santaló inequality (see, e.g., [51,Section 10.7]), an analog of Petty's inequality for the volume of projection bodies (as opposed to that of polar projection bodies) would provide an even stronger affine isoperimetric inequality.…”

“…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.…”

Blaschke-Santaló inequalities for Minkowski and Asplund endomorphisms

“…The natural question to what degree inequalities (1.2), (1.3), and (1.4) can be unified, was first asked by Lutwak. A partial answer was given in [10], deduced from results in [26], where (1.2) and (1.4) were identified as part of a larger family of inequalities for a subcone of Minkowski endomorphisms which are monotone, that is, K ⊆ L implies ΦK ⊆ ΦL for all K, L ∈ K n . For a more precise statement we require the following classification of monotone Minkowski endomorphisms.…”

“…Let us emphasize that Theorem 1 not only includes inequalities (1.2), (1.3), and (1.4) as special cases, but provides an extension of the isoperimetric inequalities from [10] from a nowhere dense set of Minkowski endomorphisms to all monotone ones. Whereas the proof of Theorem 1 does not require any results from [26], our approach is very much inspired by techniques from [26] and relies on Kiderlen's classification of monotone Minkowski endomorphisms.…”

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.…”