Timeline for Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger?
Current License: CC BY-SA 4.0
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| Dec 14, 2021 at 15:52 | comment | added | M. Winter | I definitely see the intuition behind your argument, but I also believe that it leaves room for a lot of subtleties. For example, I would need to show, that starting at $P$, I can reach any closed chain by a continuous transition that is non-increasing on all edge-lengths. This is not true if the target chain has winding number $\le 0$. You can argue this away of course (but you somewhere need to use that the circumcenter is inside the convex hull). | |
| Dec 14, 2021 at 14:27 | comment | added | Joseph O'Rourke | @M.Winter: Let the radius for the original $P$ be $r$. If you shorten any link, the polygonal chain will not close at $r$, but instead will have to shrink to a smaller radius to reach closure. | |
| Dec 14, 2021 at 13:51 | comment | added | M. Winter | Thanks for your answer. Can you explain a bit more how this relates to the proof for the case $n=m=2$? | |
| Dec 14, 2021 at 1:30 | history | answered | Joseph O'Rourke | CC BY-SA 4.0 |