Shortest curves in proximally smooth sets: existence and uniqueness
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| Format: | Preprint |
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2023
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| _version_ | 1866913584026484736 |
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| author | Ivanov, Grigory M. Lopushanski, Mariana S. Ivanov, Grigorii E. |
| author_facet | Ivanov, Grigory M. Lopushanski, Mariana S. Ivanov, Grigorii E. |
| contents | We study shortest curves in proximally smooth subsets of a Hilbert space. We consider an $R$-proximally smooth set $A$ in a Hilbert space with points $a$ and $b$ satisfying $\left|{a-b}\right| < 2R.$ We provide a simple geometric algorithm of constructing a curve inside $A$ connecting $a$ and $b$ whose length is at most $2R \arcsin\frac{\left|{a-b}\right|}{2R},$ which corresponds to the shortest curve inside the model space -- a Euclidean sphere of radius $R$ passing through $a$ and $b.$ Using this construction, we show that there exists a unique shortest curve inside $A$ connecting $a$ and $b.$ This result is tight since two points of $A$ at distance $2R$ are not necessarily connected in $A;$ the bound on the length cannot be improved since the equality is attained on the Euclidean sphere of radius $R.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_15279 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Shortest curves in proximally smooth sets: existence and uniqueness Ivanov, Grigory M. Lopushanski, Mariana S. Ivanov, Grigorii E. Functional Analysis 49J52 (primary), 49J27, 49J53 We study shortest curves in proximally smooth subsets of a Hilbert space. We consider an $R$-proximally smooth set $A$ in a Hilbert space with points $a$ and $b$ satisfying $\left|{a-b}\right| < 2R.$ We provide a simple geometric algorithm of constructing a curve inside $A$ connecting $a$ and $b$ whose length is at most $2R \arcsin\frac{\left|{a-b}\right|}{2R},$ which corresponds to the shortest curve inside the model space -- a Euclidean sphere of radius $R$ passing through $a$ and $b.$ Using this construction, we show that there exists a unique shortest curve inside $A$ connecting $a$ and $b.$ This result is tight since two points of $A$ at distance $2R$ are not necessarily connected in $A;$ the bound on the length cannot be improved since the equality is attained on the Euclidean sphere of radius $R.$ |
| title | Shortest curves in proximally smooth sets: existence and uniqueness |
| topic | Functional Analysis 49J52 (primary), 49J27, 49J53 |
| url | https://arxiv.org/abs/2308.15279 |