# 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..