The double cover relative to a convex domain and the relative isoperimetric inequality

Choe, Jaigyoung

doi:10.1017/s1446788700014087

Cited by 3 publications

(3 citation statements)

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“…If κ > 0, suppose also that chord(M) < π/ √ κ. Then M is LCD(κ) with respect to geodesics that reflect from ∂ M. Proposition 5.8 generalizes Lemma 3.2 of Choe [Cho06], which claims Candle(0) using (in the proof) the same hypotheses when κ = 0. However, the argument given there omits many details about reflection from a convex surface.…”

Section: Mirrors and Multiple Imagessupporting

confidence: 51%

“…Relative inequalities and multiple images. Choe[Cho03,Cho06] generalizes Weil's and Croke's theorems in dimensions 2 and 4 to a domain Ω ⊆ M which is outside of a convex domain C, which is allowed to share part of its boundary with ∂C; he then minimizes the boundary volume |∂ Ω ∂C|. The optimum in both cases is half of a Euclidean ball.…”

mentioning

confidence: 99%

“…Theorem 1.14 (Choe [Cho03,Cho06]). Let M be a Cartan-Hadamard manifold of dimension n ∈ {2, 4}, and let Ω ⊆ M be a domain whose interior is disjoint from a convex domain C ⊆ M. Then…”

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confidence: 99%

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The Cartan–Hadamard conjecture and the Little Prince

Kloeckner

Kuperberg

2019

Rev. Mat. Iberoam.

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The generalized Cartan-Hadamard conjecture says that if Ω is a domain with fixed volume in a complete, simply connected Riemannian n-manifold M with sectional curvature K κ 0, then ∂ Ω has the least possible boundary volume when Ω is a round n-ball with constant curvature K = κ. The case n = 2 and κ = 0 is an old result of Weil. We give a unified proof of this conjecture in dimensions n = 2 and n = 4 when κ = 0, and a special case of the conjecture for κ < 0 and a version for κ > 0. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for n = 4 and κ = 0. The generalization to n = 4 and κ = 0 is a new result. As Croke implicitly did, we relax the curvature condition K κ to a weaker candle condition Candle(κ) or LCD(κ).We also find counterexamples to a naïve version of the Cartan-Hadamard conjecture: For every ε > 0, there is a Riemannian Ω ∼ = B 3 with (1 − ε)-pinched negative curvature, and with |∂ Ω| bounded by a function of ε and |Ω| arbitrarily large.We begin with a pointwise isoperimetric problem called "the problem of the Little Prince." Its proof becomes part of the more general method.

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Section: Mirrors and Multiple Imagessupporting

confidence: 51%

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confidence: 99%