Denote the unit ball for the $p$-norm in $\mathbb{R}^N$ with $p \in (0,1]$, $$S_p^N = \Big \{ x \in \mathbb{R}^N,\ \Big(\sum \limits_{i=1}^N |x_i|^p\Big)^{1/p} \le 1 \Big\}$$
We want to find a convex subset of this ball having maximum Lebesgue measure.
My conjecture is that this set is the biggest ball for the $1$-norm that fits inside $S_p^N$. Solving $\lambda S_1^N \subset S_p^N$ yields $\lambda \le N^{1-1/p}$. The points of intersection of $S_p^N$ and $N^{1-1/p}S_1^N$ are the all the points $\big(\pm N^{-1/p},...,\pm N^{-1/p}\big)$. The Lebesgue measure of this convex set is $\frac{2^N}{N!}N^{N\big(1-\frac{1}{p}\big)}$. Is that the highest volume of a convex set inside $S_p^N$?
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For example, in $\mathbb{R}^2$, we have the figure just above. I can prove that among all losanges inside $S_{0.5}^2$, $\frac{1}{2}S_1^2$ has the highest volume, yet I can not prove that there are no other convex set that could be better, and I can't either generalize to higher dimensions.
from Hot Weekly Questions - Mathematics Stack Exchange
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