For fixed m, the rescaled volume of a random m-dimensional projection or section of an ell_p ball satisfies a CLT, an MDP, and an LDP with explicit limits as N grows.
Artstein-Avidan, A
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Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls
For fixed m, the rescaled volume of a random m-dimensional projection or section of an ell_p ball satisfies a CLT, an MDP, and an LDP with explicit limits as N grows.