Timeline for Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 27, 2022 at 11:20 | vote | accept | math110 | ||
| Mar 9, 2021 at 7:41 | comment | added | Mateusz Kwaśnicki | If $X$ is centered at the median, the last question is partially answered in item 15 of Complements and details to Section 19 of Loéve's book Probability Theory I, DOI:10.1007/978-1-4684-9464-8. He gives an inequality with an additional factor $\tfrac{(2k+1)!}{(2^kk!)^2}$ if $n=2k+1$ or $n=2k+2$. An example that follows shows that if $X=1$ with probability $\tfrac23$ and $X=-2$ with probability $\tfrac13$, then $E|X+Y+Z|=E|X+Y|=\tfrac43 E|X|$. I do not know right away if these constants are optimal. | |
| Feb 6, 2021 at 13:24 | history | edited | fedja | CC BY-SA 4.0 |
added 2166 characters in body
|
| Feb 5, 2021 at 22:42 | history | answered | fedja | CC BY-SA 4.0 |