Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed volume of $A$ and $B$ are defined by

\begin{align} MV(A,B) = \lim_{\epsilon \rightarrow 0+} \frac{ \gamma( A \oplus \epsilon \cdot B) - \gamma(A) }{\epsilon}, \end{align} where $\oplus$ denotes Minkovski addition of two sets.

It is known that if $B= \mathbb{B}_2^n$, which is the Euclidean $\ell_2$-ball in $\mathbb{R}^n$, we have \begin{align} MV(A, \mathbb{B}_2^n) = \int_{ \partial A} \mathrm{d} \sigma_{\gamma}, \end{align} where $\partial A$ is the surface of $A$ and $\mathrm{d} \sigma_{\gamma}$ is the surface measure induced by $\gamma$.

I wonder how to compute the mixed volume with respect to $\ell_p$ balls $MV(A, \mathbb{B}_{p}^n) $ in general. Here $p \in [1, \infty]$.

This problem is connected to the anti-concentration problems in probability. In http://arxiv.org/abs/1301.4807 it is shown that for $A $ to be the intersection of linear hyperplanes, we have $MV(A, \mathbb{B}_{\infty}^n) = MV(A, \mathbb{B}_2^n)$. But their derivation is not geometric and may not be able to extend to the general case.

  • $\begingroup$ Could you elaborate on what you mean by "compute the mixed volume"? There is a standard formula in convex geometry for the mixed volume of two convex bodies as an integral, and this formula applies to the case where the second body is an $\ell_p$ ball. $\endgroup$ – Deane Yang Jun 13 '16 at 6:07
  • $\begingroup$ Presumably the OP is looking for a formula that would look something like $MV(A,\mathbb B_p^n)=\int_{\partial A}f(n)\,d\sigma_\gamma$, where $f(n)$ is a function of the normal to $A$. $\endgroup$ – Anthony Quas Jun 13 '16 at 7:29

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