The author gives an answer to the question on the average complexity of a solution method for canonical linear programming of the type: maximize \(v^Tx\) subject to \(a_i^Tx\leq 1\), \(i=1,2,\dots,m\), where \(x,v,a_i\) \((1=1,2, \dots,m) \in\mathbb{R}^n\) and \(m\geq n\). The author assumes a distribution of the linear programming problems corresponding to the rotation symmetry model: the vectors \(a_1,\dots,m\), \(v\) are distributed on \(\mathbb{R}^n \setminus \{0\}\) independently, identically and symmetrically under rotations. When the polyhedron \(X=\{x\mid a_i^T\leq 1\), \(i=1,2, \dots,m\} \subset\mathbb{R}^n\) is orthogonally projected on the two-dimensional plane \(\text{lin} (u,v)\) which is a linear line of \(u\) and \(v\), then some of its vertices are mapped onto the vertices of the two-dimensional image of \(X\), where a set \(\{a_1,\dots, a_m,u, v\} \subset \mathbb{R}^n\) satisfies a mild nondegeneracy condition. These vertices of \(X\) are called shadow vertices under that projection and their number is denoted by \(S\).
The same author showed in [Z. Oper. Res., Ser. A 26, 157-177 (1982; Zbl 0488.90047) and “The simplex method, a probabilistic analysis” (1987; Zbl 0604.90092)] that there is an upper bound for the expected value \(E{m,n}(S)\) of the form \[ E_{m,n}(S)\leq m^{1/(n-1)}\cdot n^3 \cdot\text{Const} \] for all \(m\geq n\) and for all rotation symmetric distributions. In this paper the author improves the upper bounds for \(E_{m,n} (S)\) in the sharp form \[ m^{1/(n-1)} \cdot n^2\cdot \text{Const}. \]