Timeline for Distributing points with respect to a concave function
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 4, 2011 at 2:42 | answer | added | Noam D. Elkies | timeline score: 2 | |
| Sep 3, 2011 at 20:33 | comment | added | Noam D. Elkies | Is it easy to outline the proof of $F(x_1) \geq \alpha/6$? That might give a start towards the more general problem you ask. | |
| Sep 3, 2011 at 20:31 | comment | added | Noam D. Elkies | The size of $\alpha = \int_0^1 f(t) \phantom. dt$ is irrelevant: multiplying $f$ by a scalar $c$ preserves concavity and multiplies $F(x_1,\ldots,x_n)$ by the same $c$, so if a bound like $\alpha/(6n)$ holds for "small" $\alpha$ then it also holds without any such hypothesis. | |
| Aug 20, 2011 at 18:35 | answer | added | Trenton Osborn | timeline score: 1 | |
| Jun 10, 2011 at 17:26 | comment | added | Pietro Majer | Note that for a fixed $f$, the function $F$ is $C^1$ in the variables $(x_1,\dots,x_n)$, even if $f$ is only integrable. There is a minimizer on the closed symplex {$0\le x_1 \le \dots\le x_n\le 1$} by compactness, and it verifies $0 < x_1 < \dots < x_n < 1$ provided $f$ is a.e. positive. | |
| Jun 10, 2011 at 9:11 | comment | added | Dirk | Did you try to minimize $F$, e.g. by calculating its (sub-)derivatives with respect to all coordinates $x_i$ all solving the resulting equations? | |
| Jun 9, 2011 at 20:01 | history | asked | Jennifer Gao | CC BY-SA 3.0 |