Timeline for "Paradoxes" in $\mathbb{R}^n$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 1, 2021 at 17:08 | comment | added | Will Orrick | ... shell), it is not always true that the bulk is a Euclidean ball (or shell); a counterexample is the unit cube $[-1,1]^n$. In fact, the cube is the worst convex set in the Dvoretzky theorem..." So I suspect the ball of radius $\sqrt{n/3}+\epsilon$ is probably the ball that is meant. My own meager attempt to form an intuitive picture of this "ball" that is actually the intersection of a ball and a cube is here. | |
| Jan 1, 2021 at 14:06 | comment | added | Will Orrick | @TimCampion, Dirk: The link to Roman Vershynin's slides no longer works, but the same author's Estimation in high dimensions: a geometric perspective contains the same picture and similar text to the extract from the slides. Footnote 1 on page 8 clears up a few things about the round ball plus outliers picture: "This intuition is a good approximation to truth, but it should to be corrected. While concentration of volume tells us that the bulk is contained in a certain Euclidean ball (and even in a thin spherical... | |
| Dec 30, 2020 at 22:34 | comment | added | Will Orrick | @TimCampion, Dirk:I'm still a bit confused. First, isn't the limit of $\|m-e_k\|$ equal to 1, not $\infty$? It is true that the ratio of the circumradius to the inradius approaches $\infty$, but not the circumradius itself. Second, the ratio of the volume of the inscribed ball to that of the simplex approaches 0 (just as for the $n$-cube), so what is the ball in Milman's picture? I'm aware of the result that most of the volume of the cube $[-1,1]^n$ lies in a spherical shell of radius $\sqrt{n/3}$, but this sphere does not lie entirely within the cube, so I'm not sure whether this is relevant. | |
| Sep 8, 2014 at 13:48 | comment | added | Dirk | The unit ball is indeed a special convex body. | |
| Sep 8, 2014 at 1:47 | comment | added | Tim Campion | The idea that the bulk has little volume seems to contradict Nik Weaver's point that most of the volume of the unit ball is concentrated near the boundary... It also seems to be in tension with the fact that the unit sphere takes up a vanishing fraction of the volume of the unit cube. What gives? | |
| Mar 13, 2014 at 6:32 | comment | added | Alexander Shamov | Incidentally, that could be a hyperbolic picture as well... | |
| Mar 10, 2014 at 11:28 | comment | added | Campello | Sorry... whereas the distance to the corner is $\sqrt{n}/2$, that goes to $\infty$. | |
| Mar 10, 2014 at 11:20 | comment | added | Dirk | Right - the cube is actually a simpler example than the simplex... | |
| Mar 10, 2014 at 11:19 | comment | added | Campello | The same happens to the cube. The midpoint (1/2,...1/2) has distance 1/2 do any of the faces, whereas the distance to the corner is $1$. Thanks for the slides. | |
| Mar 10, 2014 at 10:16 | history | answered | Dirk | CC BY-SA 3.0 |