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In this note, we assess the accuracy of CLT-based approximations for thevolume of intersection of the $d$-dimensional cube $[-1,1]^d$ and an $L_q$-ballcentred at the origin; this is clearly equivalent to approximating thedistribution of the $L_q$-norm of a random point in a $d$-dimensional cubecentered at 0. The approximations are CLT-based where to improve the normalapproximation we use the first term in the Edgeworth expansion.  We have included a section analysing the information obtained from ChatGPT inresponse to prompts regarding this theory; in our case, ChatGPT answers werenot very helpful. Illustrations of the approximation formulae, as the radius ofthe ball increases, for different values of $d$ and $q$ are also given,alongside lines showing a Monte Carlo simulation of the intersection volume.

Approximating the distribution of the $L_q$-norm of a random point in a $d$-dimensional cube