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arXiv · 1907.09067

A non-geodesic analogue of Reshetnyak's majorization theorem

Tetsu Toyoda
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Abstract

For any real number $κ$ and any integer $n\geq 4$, the $\mathrm{Cycl}_n (κ)$ condition introduced by Gromov (2001) is a necessary condition for a metric space to admit an isometric embedding into a $\mathrm{CAT}(κ)$ space. It is known that for geodesic metric spaces, the $\mathrm{Cycl}_4 (κ)$ condition is equivalent to being $\mathrm{CAT}(κ)$. In this paper, we prove an analogue of Reshetnyak's majorization theorem for (possibly non-geodesic) metric spaces that satisfy the $\mathrm{Cycl}_4 (κ)$ condition. It follows from our result that for general metric spaces, the $\mathrm{Cycl}_4 (κ)$ condition implies the $\mathrm{Cycl}_n (κ)$ conditions for all integers $n\geq 5$, although Gromov stated that this implication is apparently not true.

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Tetsu Toyoda. 2022-11-30. A non-geodesic analogue of Reshetnyak's majorization theorem. https://arxiv.org/abs/1907.09067

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