Timeline for Planar sections of convex sets in Cartan-Hadamard manifolds
Current License: CC BY-SA 4.0
14 events
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| Oct 29, 2023 at 1:39 | comment | added | Mohammad Ghomi | @MoisheKohan: The induced metric. | |
| Oct 29, 2023 at 0:47 | comment | added | Moishe Kohan | What metric do you put on $\Pi$? These submanifolds are hardly ever totally geodesic and, thus, will intersect geodesic segments in $X$ in disconnected subsets. I do not see how they can be possibly used to detect convexity in $X$ unless $X$ has constant curvature. | |
| Oct 29, 2023 at 0:46 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Oct 29, 2023 at 0:08 | comment | added | Mohammad Ghomi | @IgorBelegradek: Yes, I changed the word "characterized" to "identified". | |
| Oct 29, 2023 at 0:06 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Oct 28, 2023 at 23:53 | comment | added | Igor Belegradek | So you are not asking for a characterization. It is one way: if each $\Pi\cap X$ is convex, then $X$ is convex. | |
| Oct 28, 2023 at 23:33 | comment | added | Mohammad Ghomi | @IgorBelegradek: The question is whether $X$ is convex assuming that $\Pi\cap X$ is a convex subset of $\Pi$, for all planes $\Pi$ passing through some point $p$ of $M$. A subset of $\Pi$ (which itself is a Cartan-Hadamard space) is convex if every pair of points of $\Pi$ can be joined by a geodesic of $\Pi$ which lie in that set. | |
| Oct 28, 2023 at 22:54 | comment | added | Igor Belegradek | Sorry, I still don't understand the proposed characterization. | |
| Oct 28, 2023 at 22:06 | comment | added | Mohammad Ghomi | @IgorBelegradek: Thanks, I edited the question to clarify this point. | |
| Oct 28, 2023 at 22:04 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Oct 28, 2023 at 20:50 | comment | added | Igor Belegradek | Are you asking whether one can characterize convex subsets of Cartan-Hadamard manifold in terms of their two-dimensional "sections"? What is meant by a "section" in your question? As you say, in a general Cartan-Hadamard manifold there will be no 2-dimensional convex set whose relative interior contains a given point, so we cannot define a "section" as a 2-dimensional convex set. | |
| Oct 28, 2023 at 17:09 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Oct 28, 2023 at 16:28 | history | edited | Mohammad Ghomi | CC BY-SA 4.0 |
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| Oct 28, 2023 at 16:17 | history | asked | Mohammad Ghomi | CC BY-SA 4.0 |