Timeline for Unusual isoperimetry and maximizing the measure of unions of translates of a set
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 19, 2015 at 21:46 | comment | added | Bruno Le Floch | @TOM I'm changing $A$ to be some union of intervals fattened by $\epsilon$. I should have called it $B$ probably to limit confusion. So $B$ is essentially the image of Anton Petrunin's $A$ under your operation. Acting a second time with your operation will not change the volume much. | |
| Nov 19, 2015 at 21:44 | comment | added | TOM | Just to make thing clearer - I assumed that $\varepsilon$ is the volume of one little ball and not the radius. | |
| Nov 19, 2015 at 21:41 | comment | added | TOM | @BrunoLeFloch : I do not quite agree, the unions of shifts of one ball by a line segment of length $l$ will have length $\approx l\varepsilon$ and the whole union of shifts $\approx 2ldN^2\varepsilon$, which is quite different from the measure of $A$, which is like $N^2\varepsilon$. Or am I saying something silly? | |
| Nov 19, 2015 at 19:59 | comment | added | Bruno Le Floch | @TOM: still, take $A$ to be the union of $\epsilon$-balls whose centers have all coordinates integer except one, all coordinates between $\pm N$. In other words the centers are along edges between neighboring integer-coordinate points considered by Anton Petrunin. Then the shifted $A$ will essentially have the same form, but with $N\to N+1$, and again there is no minimizer. | |
| Nov 19, 2015 at 19:01 | comment | added | TOM | Perhaps the "closer" way of modifying the question is the following. So by adding a ball to the set I extend it by a segment of a fixed length in every direction. What about changing the discrete shifts to shifts in the same 2d directions by a segment of fixed length? That seems to at least kill your couterexample for the discrete question. | |
| Nov 19, 2015 at 18:37 | comment | added | Anton Petrunin | @TOM, things might change if you assume convexity. | |
| Nov 19, 2015 at 16:48 | comment | added | TOM | Thank you! I guess then one cannot get away with finitely many directions for shifts and get a non-trivial answer. | |
| Nov 19, 2015 at 16:46 | vote | accept | TOM | ||
| Nov 19, 2015 at 15:58 | history | answered | Anton Petrunin | CC BY-SA 3.0 |