Fixed points of Minkowski valuations

Ortega-Moreno, Oscar; Schuster, Franz E.

doi:10.1016/j.aim.2021.108017

Cited by 2 publications

(6 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Section 5.2 is dedicated to Minkowski valuations Φ i generated by origin-symmetric convex bodies of revolution. There we prove Theorem A, unifying previous results by Ivaki [29,30] and the second author and Schuster [48]. Finally, in Section 5.3 we consider general even Minkowski valuations for which we obtain information about the fixed points of Φ i (as opposed to Φ 2 i ).…”

Section: Fixed Pointssupporting

confidence: 69%

“…The case when i = 1 (that is, Φ 1 ∈ MVal 1 is a Minkowski endomorphism) has been settled globally by Kiderlen [32]. With Theorem A, we unify the previous results on C 2 + regular Minkowski valuations and projection body maps obtained in [48] and [29,30], respectively. However, none of them (including Theorem A) cover any local uniqueness of fixed points of mean section operators.…”

Section: Introductionsupporting

confidence: 53%

“…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%

“…Remark 5.3. The way Theorem 5.1 was stated in [48] additionally required Φ i to be even as the proof employs the following classical result by Strichartz [62]. Denote by H s (S n−1 ), s ∈ N, the Sobolev space of functions on S n−1 with weak covariant derivatives up to order s in L 2 (S n−1 ).…”

Section: Fixed Pointsmentioning

confidence: 99%

“…In [48], this lemma was proven only for C 2 [−1, 1] functions, however it extends to C[−1, 1] by a simple approximation argument. Next, we determine the kernels of A 1 and A 2 .…”

Section: Convex Bodies Of Revolutionmentioning

confidence: 99%

See 4 more Smart Citations

Fixed Points of Mean Section Operators

Brauner¹,

Ortega-Moreno²

2023

Preprint

View full text Add to dashboard Cite

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

show abstract

Section: Fixed Pointssupporting

confidence: 69%

Section: Introductionsupporting

confidence: 53%

Section: Introductionmentioning

confidence: 89%

Section: Fixed Pointsmentioning

confidence: 99%

“…In [48], this lemma was proven only for C 2 [−1, 1] functions, however it extends to C[−1, 1] by a simple approximation argument. Next, we determine the kernels of A 1 and A 2 .…”

Section: Convex Bodies Of Revolutionmentioning

confidence: 99%

See 3 more Smart Citations