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We establish a family of isoperimetric inequalities for sets that interpolate between intersection bodies and dualL_{p}-centroid bodies. This provides a bridge between the Busemann intersection inequality and the Lutwak–Zhang inequality. The approach depends on new empirical versions of these inequalities.
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Affine isoperimetric inequalities for higher-order projection and centroid bodies
Abstract In 1970, Schneider introduced the mth order difference body of a convex body, and also established the mth-order Rogers–Shephard inequality. In this paper, we extend this idea to the projection body, centroid body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang’s projection inequality, Petty’s projection inequality, the Busemann–Petty centroid inequality and Busemann’s random simplex inequality). We also establish a new proof of Schneider’s mth-order Rogers–Shephard inequality. As an application, a mth-order affine Sobolev inequality for functions of bounded variation is provided.
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- Award ID(s):
- 1929284
- PAR ID:
- 10631679
- Publisher / Repository:
- Springer Nature Link
- Date Published:
- Journal Name:
- Mathematische Annalen
- ISSN:
- 0025-5831
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1007/s00208-025-03271-x
