Timeline for Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 29, 2021 at 17:48 | comment | added | dohmatob | BTW, all these questions and answers about Lipschitzness of normal fields have helped me build a solution to a bigger problem mathoverflow.net/a/412651/78539. Thanks again. | |
| Dec 29, 2021 at 16:54 | comment | added | dohmatob | Yes, that's that's the example I was referring to; forgot to link it. | |
| Dec 29, 2021 at 16:50 | comment | added | Dustin G. Mixon | Indeed, the necessary condition rigorizes my comment here: mathoverflow.net/questions/412663/… | |
| Dec 29, 2021 at 16:31 | comment | added | dohmatob | Ok, in that case, your necessary condition would explain why a polytope with at least one face of positive-codimension fails to have Lipschitz $u_C$. This is particularly easy to see in the pathological case where $C=\{0\}$, for example. Indeed, one computes $(C-0)^\circ = \{0\}^\circ = \mathbb R^n$ which is $n$-dimensional! Generalizing to other polytopes might be more involved but should follow the same spirit. Thanks again. | |
| Dec 29, 2021 at 16:23 | comment | added | Dustin G. Mixon | Yep, that's what I have in mind. | |
| Dec 29, 2021 at 16:22 | comment | added | dohmatob | Thanks for your insightful solution. The sufficient condition (2) is particular satisfying since this one of the cases I really care about. BTW, since there is not natural definition of "dimension of polar cone", what do you have in mind ? Dimension of smallest affine subset of $\mathbb R^n$ which contains the polar cone ? | |
| Dec 29, 2021 at 16:20 | vote | accept | dohmatob | ||
| Dec 29, 2021 at 16:13 | history | edited | Dustin G. Mixon | CC BY-SA 4.0 |
added 768 characters in body
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| Dec 29, 2021 at 15:35 | history | answered | Dustin G. Mixon | CC BY-SA 4.0 |