Average number of solutions for systems of equations

For $n$ finite-dimensional spaces of smooth functions $V _i $ on a smooth $n$-dimensional manifold $X$, the systems of equations $ \{f_i = a_i \colon \: f_i \in V_i, \: a_i \in \mathbb{R}, \: i = 1, \ldots, n \} $ are considered. A connection is established between the average numbers of solutions and the mixed volumes of convex bodies. To do this, fixing Banach metrics of the spaces $ V_i $, we construct 1) measures in the spaces of systems of equations, and 2) Banach convex bodies in $X$, those. families of centrally symmetric convex bodies in the layers of the cotangent bundle $X$. It is proved that the average number of solutions is equal to the mixed symplectic volume of Banach convex bodies. The case of Euclidean metrics in the spaces $ V_i $ was previously considered. In this case, the Banach bodies are ellipsoid families.