How to calculate the volume of a section of a convex body?

The following is essentially a partial case for my previous question.

Let $ B\subset\mathbb{R}^m$ be the unit ball with respect to a concrete norm on $ \mathbb{R}^m$ , say $ l^p$ -norm, $ p\in (1,\infty)$ . Let $ v_1,…,v_n\in \mathbb{R}^m$ be linearly independent.

How to calculate the $ n$ -dimensional volume of $ B\cap span\{v_1,…,v_n\}$ ?

I need to express this volume through the coordinates of $ v_1,…,v_n$ , or perhaps through some distances between certain combinations of them. I know that there is extensive literature on related matters, but I hope that this specific question has a specific answer..