Timeline for Strange result about convexity
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 1, 2021 at 14:05 | comment | added | pinaki | Thanks for the valuable insight about convexity and divided differences - this is one of the best answers I came across on Mathoverflow. | |
| Oct 31, 2021 at 14:25 | comment | added | zeb | @Iosif The function $f(x) = x^k$ can indeed be replaced by any given $(k-1)$th-order strictly convex function - the only purpose of it is to make sure that you are getting the direction of the inequality right! Farkas lemma is unnecessary here, all you need to do is to check that the system of linear equations (satisfied by the $c_i$) which you get by checking that you have equality at $f(x) = x^m$ for $m < k$ is non-degenerate, so that it has just one solution, up to scale. | |
| Oct 31, 2021 at 11:57 | comment | added | Iosif Pinelis | I am wondering if your theorem can be deduced from the Farkas lemma (en.wikipedia.org/wiki/Farkas%27_lemma) and/or using the generalized Vandermonde matrix (jstor.org/stable/2690290). Also, wondering if $f(x)=x^k$ in your theorem can be replaced by $f(x)=e^x$ or, more generally, by $f(x)=g(x)$, where $g$ is any given $(k−1)$th-order strictly convex function. | |
| Oct 31, 2021 at 1:39 | history | edited | LSpice | CC BY-SA 4.0 |
TeX fix; name of article
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| Oct 30, 2021 at 20:31 | history | answered | zeb | CC BY-SA 4.0 |