Timeline for Is the max-centre map continuous for open bounded domains?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2022 at 16:19 | comment | added | JHM | Applying your argument verbatim appears to imply the nonexistence of any strong deformation retracts from $A$ onto $M(A)$. So Blum's Medial Axis Theorem, which claims a homotopy-isomorphism $A \simeq M(A)$, appears to not be obtained by a strong deformation retract for pathological $A$ like your example above. But if $A$ has no small-scale variations, then the strong retract is possible, and I'm arguing the max-centre map is the strong retract $A \leadsto M(A)$. | |
| Jan 10, 2022 at 20:13 | comment | added | JHM | It's something about Cauchy sequences in the background euclidean distance which are not Cauchy sequences in the subset metric. In your bean example, the branches accumulate w.r.t. euclidean distance but are not Cauchy in the subset metric (as measured in the medial axis along the branches). This is some kind of metric condition on the inclusion $M(A)\hookrightarrow A$. | |
| Jan 10, 2022 at 19:53 | comment | added | Saúl RM | Hm but in the set $\{(x,y)\in\mathbb{R}^2;10>y>x^2;\}$, the medial axis is not closed in $A$ (the point $(0,0.5)$ is in the closure) but $m$ seems to be continuous | |
| Jan 10, 2022 at 19:30 | comment | added | JHM | The medial axis is like infinite trivalent tree whose branches accumulate to the vertical line segment $(0,t), 0\leq t \leq 1/2$ in the plane. The points $p_n$ belong to the distinct "branches". So you've identified the fact that $m$ is discontinuous whenever the medial axis is not relatively closed in $A$. | |
| Jan 10, 2022 at 18:48 | comment | added | JHM | Interesting bean example! Thank you. | |
| Jan 10, 2022 at 17:41 | history | edited | user44143 | CC BY-SA 4.0 |
simplified
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| Jan 10, 2022 at 17:33 | comment | added | Saúl RM | I don´t know what condition could work. There are convex counterexamples (consider the disk $D(0,1)$ and the points $e^\frac{i}{n}$ in its boundary, and the tangent lines at those points. Then the convex region delimited by those lines and the disk, (the disk but adjoining a sequence of small "triangles"), seems to be a counterexample too). Maybe some strong differentiability condition on the boundary could work. Also, my example isn´t stable, if by stable you mean under small changes (small in Hausdorff distance) or sth like that. Have fun with the medial axis of the bean :D | |
| Jan 10, 2022 at 16:52 | vote | accept | JHM | ||
| Jan 10, 2022 at 15:30 | history | edited | Saúl RM | CC BY-SA 4.0 |
added 13 characters in body
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| Jan 10, 2022 at 15:23 | history | answered | Saúl RM | CC BY-SA 4.0 |