Abstract:
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.
Keywords:Banach space, Crofton formula, normal density, mixed volume.
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