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An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space R d \mathbb {R}^d of specified Lebesgue measures, (tuples of) balls centered at the origin are maximizers of certain functionals defined by multidimensional integrals. For d > 1 d>1 , this inequality only applies to functionals invariant under a diagonal action of Sl ( d ) \operatorname {Sl}(d) . We investigate functionals of this type, and their maximizers, in perhaps the simplest situation in which Sl ( d ) \operatorname {Sl}(d) invariance does not hold. Assuming a more limited symmetry encompassing dilations but not rotations, we show under natural hypotheses that maximizers exist, and, moreover, that there exist distinguished maximizers whose structure reflects this limited symmetry. For small perturbations of the Sl ( d ) \operatorname {Sl}(d) –invariant framework we show that these distinguished maximizers are strongly convex sets with infinitely differentiable boundaries. It is shown that in the absence of partial symmetry, maximizers fail to exist for certain arbitrarily small perturbations of Sl ( d ) \operatorname {Sl}(d) –invariant structures.
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This content will become publicly available on March 3, 2026
Two results on the Convex Algebraic Geometry of sets with continuous symmetries
Abstract We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral description, that is, can be described by an equivariant linear matrix inequality. Second, we show that the bijection induced by Kostant's Convexity Theorem between convex subsets invariant under a polar representation and convex subsets of a section invariant under the Weyl group preserves the classes of convex semialgebraic sets, spectrahedral shadows, and rigidly convex sets.
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- Award ID(s):
- 2142575
- PAR ID:
- 10576500
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 57
- Issue:
- 5
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 1388-1408
- Size(s):
- p. 1388-1408
- Sponsoring Org:
- National Science Foundation
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