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I want to compute the value of the following integral: $$\int_{r\in\mathbb{R}^d: \|\|r\|\|\le 1} \exp(a^Tr)dr$$.

In particular, $a$ is the coefficient and the norm $\|\|\cdot\|\|$ could be the general norm function instead of $2$-norm.

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  • $\begingroup$ For the 2-norm this is answered (in cyrillic) in Dima Pasechnik’s answer to this question. For general norm, no idea. $\endgroup$
    – Francois Ziegler
    Commented Feb 2, 2023 at 19:47
  • $\begingroup$ the $L_1$ norm result is given here mathoverflow.net/a/416254/11260 --- perhaps that calculation can be adapted to $L_p$ norms, something to check. $\endgroup$
    – Carlo Beenakker
    Commented Feb 2, 2023 at 22:11

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