Timeline for Approximating a convex disk by an ellipse
Current License: CC BY-SA 3.0
32 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Feb 16, 2017 at 18:40 | comment | added | Wlodek Kuperberg | @YaakovBaruch: I don't think it is obvious no matter which way it goes. So, If you find a proof that for some symmetric trapezoid the optimal ellipse is not aligned with the bases, please write it in as an answer. By the way, the idea of using symmetry for finding a counterexample was tried already, so far with no success. | |
| Feb 16, 2017 at 9:00 | comment | added | Yaakov Baruch | @WlodekKuperberg: if an optimal ellipse is not aligned with the bases, then by symmetry there are at least 2 different optimal ellipses. I asked because, visually, it doesn't appear obvious to me that that is not the case (although I think it's unlikely). | |
| Feb 15, 2017 at 23:39 | comment | added | Wlodek Kuperberg | @YaakovBaruch: what about them? Is it known that it does not? By the way, the problem is affine-invariant, so axial symmetry is irrelevant. | |
| Feb 15, 2017 at 9:28 | comment | added | Yaakov Baruch | What about symmetric trapezoids? Is it known that the optimal ellipse has an axis parallel to the bases? Unless I missed some, I didn't see any comments mentioning trapezoids. | |
| Feb 13, 2017 at 0:01 | vote | accept | Wlodek Kuperberg | ||
| Feb 11, 2017 at 21:15 | answer | added | Jairo Bochi | timeline score: 5 | |
| Jan 4, 2017 at 5:24 | answer | added | user44143 | timeline score: 3 | |
| S Jan 4, 2017 at 5:06 | history | bounty ended | Wlodek Kuperberg | ||
| S Jan 4, 2017 at 5:06 | history | notice removed | Wlodek Kuperberg | ||
| Jan 1, 2017 at 23:42 | comment | added | Wlodek Kuperberg | In the above comment, I should have added that the metric based on the volume of the symmetric difference is topologically equivalent to the original Banach-Mazur metric. But if it happens that the ellipsoid in question is unique, then some nice geometric properties of the Banach-Mazur continuum would follow, such as its starlike shape, with the affine class of ellipsoid at the center, and straight-line segments connecting the center with all other points. (I wrote this in a separate comment because of the limit on the number of characters allowed in a comment.) | |
| Jan 1, 2017 at 23:34 | comment | added | Wlodek Kuperberg | The question is equivalent to that of an ellipse $E$ (or ellipsoid) of area $1$ whose intersection with $K$ is of maximum area, and this version seems nicer for two reasons: (1) it sounds simpler and (2) the intersection is convex. Still, there is a reason why I asked the way I did. One can define the similar minimum value of the symmetric difference for two arbitrary convex bodies $K$ and $L$, each of volume $1$, using all bodies $L'$ affinely equivalent to $L$. The remark of Jairo Bochi on the affine invariance holds here, too. This leads to another metric on the Banach-Mazur compactum. | |
| Dec 28, 2016 at 22:59 | vote | accept | Wlodek Kuperberg | ||
| Dec 28, 2016 at 23:07 | |||||
| Dec 28, 2016 at 18:14 | answer | added | Jairo Bochi | timeline score: 3 | |
| S Dec 28, 2016 at 17:32 | history | bounty started | Wlodek Kuperberg | ||
| S Dec 28, 2016 at 17:32 | history | notice added | Wlodek Kuperberg | Improve details | |
| Dec 27, 2016 at 22:57 | answer | added | Iosif Pinelis | timeline score: 5 | |
| Dec 24, 2016 at 5:42 | answer | added | Gerhard Paseman | timeline score: 7 | |
| Dec 23, 2016 at 1:21 | comment | added | Wlodek Kuperberg | @JosephO'Rourke: The pair of "Banach-Mazur" homothetic ellipsoids is relevant, too, one contained in-, the other containing $K$, with the minimum homothety coefficient -- as in the definition of the Banach-Mazur metric. But uniqueness of the pair fails already in dimension 3, even for centrally symmetric $K$. | |
| Dec 23, 2016 at 1:15 | answer | added | Joseph O'Rourke | timeline score: 11 | |
| Dec 22, 2016 at 23:19 | comment | added | Joseph O'Rourke | Besides the John-ellipsoid, I wonder if the Milman-ellipsoid is relevant? I can only find this PNAS paper on the M-ellipsoid: "An M-ellipsoid $E$ of a convex body $K$ has small covering numbers with respect to $K$." | |
| Dec 22, 2016 at 23:00 | comment | added | Wlodek Kuperberg | @T.Amdeberhan: I believe so. But I could be wrong, of course. | |
| Dec 22, 2016 at 22:40 | comment | added | T. Amdeberhan | Thanks. Replace $K$ by a square. Then, $E$ is a circle again? | |
| Dec 22, 2016 at 22:31 | comment | added | Wlodek Kuperberg | @T.Amdeberhan: For a circle $K$, $E=K$. (A circle is an ellipse, too.) | |
| Dec 22, 2016 at 22:28 | comment | added | Wlodek Kuperberg | @GerhardPaseman: I believe it is a circle, because for the (truncated) triangle each of the inscribed maximum-area ellipse and the circumscribed minimum-area ellipse is a circle. This is not a proof, but a reason to conjecture that it is so. | |
| Dec 22, 2016 at 22:25 | comment | added | T. Amdeberhan | Suppose $K$ is the actual disk on the plane (normalized to have unit area). Assume $E$ is such an ellipse. Now, if you rotate the disk around wouldn't you get another $E'$? | |
| Dec 22, 2016 at 22:12 | comment | added | Gerhard Paseman | I can imagine (but not readily prove) that there are three non circular ellipses which might serve for a truncated modification of an equilateral triangle (or perhaps no modification is needed). Can you prove that a circle would be minimal? Gerhard "Not Seeing The Numbers Yet" Paseman, 2016.12.22. | |
| Dec 22, 2016 at 22:03 | comment | added | domotorp | The first sentence follows from a standard compactness argument. Disk is also used to denote any, typically convex set. | |
| Dec 22, 2016 at 22:02 | comment | added | Wlodek Kuperberg | @T.Amdeberhan: A compact convex set of area $1$ is necessarily a convex disk (a convex $2$-dmensional body). It is known that each of, the maximum-area ellipsoid contained in a convex body $K$ in $\mathbb{R}^n$ and the minimum-area ellipsoid containing $K$ is unique. | |
| Dec 22, 2016 at 22:01 | comment | added | T. Amdeberhan | Do you have a reference for the statement you made before the questions? Also, you title saying "a disk" but you statement in the first line says 'a compact set". Which one do you want? | |
| Dec 22, 2016 at 21:59 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
added 38 characters in body
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| Dec 22, 2016 at 21:53 | history | asked | Wlodek Kuperberg | CC BY-SA 3.0 |