Iterations of Minkowski valuations

Ortega-Moreno, Oscar

doi:10.1016/j.jfa.2023.109887

Cited by 1 publication

(1 citation statement)

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“…For degree 1 < i < n − 1, Ivaki [29] showed that in some C 2 neighborhood of the unit ball, the only fixed points of Π 2 i are Euclidean balls. The second author and Schuster [47,48] have shown that this phenomenon also holds for the class of even C 2 + regular Minkowski valuations, that is, Minkowski valuations generated by the support function of an origin-symmetric convex body of revolution that has a C 2 boundary with positive Gauss curvature. A line segment is clearly not of this kind, so the results by Ivaki and by the second author and Schuster appear to be disconnected.…”

Section: Introductionmentioning

confidence: 89%

Fixed Points of Mean Section Operators

Brauner¹,

Ortega-Moreno²

2023

Preprint

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We characterize rotation equivariant bounded linear operators from C(S n−1 ) to C 2 (S n−1 ) by the mass distribution of the spherical Laplacian of their kernel function on small polar caps. Using this characterization, we show that every continuous, homogeneous, translation invariant, and rotation equivariant Minkowski valuation Φ that is weakly monotone maps the space of convex bodies with a C 2 support function into itself. As an application, we prove that if Φ is in addition even or a mean section operator, then Euclidean balls are its only fixed points in some C 2 neighborhood of the unit ball. Our approach unifies and extends previous results by Ivaki from 2017 and the second author together with Schuster from 2021.We transform both integrals to spherical cylinder coordinates. For the left hand side, we haveand for the right hand side,where the final equality is obtained from integration by parts. This yields (3.3).For the second part of the lemma, let ψ ∈ D(−1, 1) be an arbitrary test function. Due to the spherical divergence theorem for 2-tensor fields and (3.2),We transform both integrals to spherical cylinder coordinates. For the left hand side, we have

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Section: Introductionmentioning

confidence: 89%