2
$\begingroup$

Let $x\in \mathbb{R}^n$ and let $s_k(x)$ denote the sum of the $k$ largest entries of $x$. The function $s_k(x)$ is well-known to be convex and is often used in optimization, such as

https://inst.eecs.berkeley.edu/~ee127a/book/login/def_sum_largest.html

My question: is there a name for the infinite-dimensional version of this? That is, say $f(x)$ is an integrable function on $\mathbb{R}^d$, and let $S_v(f)$ be the largest possible integral of $f(x)$ over a sub-region whose volume is $v$. Obviously, the optimal sub-region would just be the interior of a level set of $f$, but this seems like it would be a natural quantity that would have a name attached to it.

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ Sorry, I believe that one of the related links tells me that this is the "highest posterior density region" associated with $f$: mathoverflow.net/questions/7857/… , so this is probably already answered. $\endgroup$
    – Terence Short
    Mar 3, 2015 at 23:17

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.