The author investigates quadratic functions \(f=f_ 0+\sum f_ ix_ i+\sum f_{ij}x_ ix_ j\), \(f_{ij}=f_{ji}\) on \(\mathbb{R}^ n\) that satisfy the additional condition \(f(z)\geq 0\), \(z\in\mathbb{Z}^ n\). A point of a cone belongs to each of these quadratic functions in the \({n+2\choose 2}\)-dimensional space of coefficients. The root figure of \(f\) is the set of \(n\)-vectors \(z\in\mathbb{Z}^ n\) satisfying the equation \(f(z)=0\). The root figures relate to the convex structure of the cone. This paper deals with some basic questions relating to the classification of the root figures up to the symmetry group of the cone.