Timeline for Metric analogues of bounded variation
Current License: CC BY-SA 3.0
16 events
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| Mar 28, 2016 at 9:06 | comment | added | Aryeh Kontorovich | Thanks again. I guess there's a reason why the Hardy-Krause extension to higher dimensions is so complicated. | |
| Mar 27, 2016 at 21:09 | comment | added | Willie Wong | The scaling is still not right in two dimensions. Again let $f(x,y) = x$ on $[0,1]\times[0,1]$. The lattice of multiples of $1/k$ forms more-or-less a $1/k$-net. For each of the squares you have $\sup f - \inf f = 1/k$. And you have a total of $k^2$ squares. Take $k\to \infty$ you still get blow-up. The $\sup f - \inf f$ definition captures what I feel more like a 1 dimension effect. Summing over a 2-dimensional (or higher) grid will always lead to blow-up. | |
| Mar 26, 2016 at 17:08 | comment | added | Aryeh Kontorovich | @WillieWong One last attempt before giving this a rest. What if we consider an $\epsilon$-net (i.e., minimal cover/maximal packing), and then take $\mathcal{P}_n$ to be the Voronoi regions induced by the net points? | |
| Mar 22, 2016 at 20:11 | comment | added | Christian Remling | Maybe a better way to go about it is to recall that $f\in BV$ if and only if (the distribution) $f'$ is a measure, and then generalize this condition. | |
| Mar 22, 2016 at 16:29 | comment | added | Willie Wong | JFYI: the paper on arXiv you linked to in your last comment, in spite of the title, actually considers metric measure spaces, and not just metric spaces. | |
| Mar 22, 2016 at 15:33 | comment | added | Aryeh Kontorovich | Very nice example, @WillieWong, and thanks Suvrit for the link. I was also pointed to arxiv.org/pdf/1301.6897v1.pdf . So I guess it's time to quite amateur conjecturing and start reading... | |
| Mar 22, 2016 at 15:31 | comment | added | Suvrit | On a related note, have you already considered the following: archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1990_4_17_3/… | |
| Mar 22, 2016 at 15:17 | comment | added | Willie Wong | For your EDIT II: let your metric space be $[0,1]\times[0,1]$ and let $f(x,y) = x$. Let $n > 1$. Take $(x_i,y_i) = (0, i / (n-1))$ for $i < n$. Then for the $\mathcal{P}_n$ adapted to $\{(x_i,y_i)\}_{i = 0, \ldots, n-1}$ you have that $$ \sum_{A\in \mathcal{P}_n} (\sup_A f - \inf_A f) = n $$ So after taking the sup in $n$ you get that this function has infinite total variation by your definition. | |
| Mar 22, 2016 at 15:13 | comment | added | Willie Wong | There is some notion of BV functions for metric measure spaces, which seems more natural when looking at the definition for BV in several real variables. Google gives a bunch of results, two of which are cvgmt.sns.it/paper/2738 and sciencedirect.com/science/article/pii/S0021782403000369 | |
| Mar 22, 2016 at 14:00 | history | edited | Aryeh Kontorovich | CC BY-SA 3.0 |
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| Mar 22, 2016 at 13:16 | comment | added | Aryeh Kontorovich | Thank you for that example, @MartinHairer. Rather than risk further embarrassment with additional edits, let me try to continue in the comments. Suppose I define $\mathcal{P}_n$ to be the Voronoi partition induced by some $n$ points, and define the variation to be the supremum over all $\mathcal{P}_n$. Does this still admit pathological examples? | |
| Mar 22, 2016 at 13:02 | comment | added | Martin Hairer | Blocks in a partition don't need to be connected. In particular, one can easily have a partition of $[0,1]$ into $1/\epsilon^2$ blocks of diameter $\epsilon$, so the identity map still has infinite variation with the new definition. There is a notion of BV for functions of several real variables (see Wikipedia), but I am not sure that it generalises well. One naive generalisation would be to ask that $f\circ g$ is BV, uniformly over all $g \colon \mathbb{R} \to X$ having Lipschitz constant $1$. | |
| Mar 22, 2016 at 12:49 | comment | added | Aryeh Kontorovich | You're of course right, @AugustCleaner! I tried to fix this in the edited version. | |
| Mar 22, 2016 at 12:48 | history | edited | Aryeh Kontorovich | CC BY-SA 3.0 |
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| Mar 22, 2016 at 10:14 | comment | added | August Cleaner | I turn your attention to the fact that the suggested notion is quite different from the classical, e.g. the function $y=x$ on $[0,1]$ has infinite variation. Also this notion does not depend on the metric. | |
| Mar 22, 2016 at 9:36 | history | asked | Aryeh Kontorovich | CC BY-SA 3.0 |