  Let $X$ be an $n$-dimensional manifold and $V_1,\ldots,V_n\subset
C^\infty(X,\mathbb R)$ finite-dimensional vector spaces. For systems of
equations $\{f_i = a_i\colon\: f_i\in V_i,\:a_i \in\mathbb R,\:i=1,\ldots,n\}$
we discover a relationship between the average number of their solutions and
mixed volumes of convex bodies. To do this, we choose Banach metrics in the
spaces $V_i$. Using these metrics, we construct 1) the measure in the space of
systems, and 2) Banach convex bodies in $X$, i.e., collections of centrally
symmetric convex bodies in the fibers of the cotangent bundle of $X$. It turns
out that the average number of solutions is equal to the mixed symplectic
volume of Banach convex bodies. Earlier this result was obtained for Euclidean
metrics in spaces $V_i$. In Euclidean case, the Banach convex bodies are the
collections of ellipsoids.
