Deriving convex hulls through lifting and projection (Q1646568)

The authors focus on the convex hulls of the subsets of a compact hypercube defined by the constraints \(x^{b_1}_1 x^{b_2}_2\geq x_3\) and \(x_1x^{b_2}_2\leq x_3,\) where \(b_1,b_2\geq 1.\) Via a lift-and-project technique, they derive the closed expression for the convex hulls of the sets \[ S^\geq :=\{ (x_1, x_2, x_3)\in [l_1, u_1]\times [l_2, u_2] \times[l_3, u_3]\big| \, x^{b_1}_1 x^{b_2}_2 \geq x_3\} \] \[ S^\leq :=\{ (x_1, x_2, x_3)\in [l_1, u_1]\times [l_2, u_2] \times[l_3, u_3]\big| \, x_1 x^{b_2}_2 \leq x_3\}, \] and \[ S^= :=\{ (x_1, x_2, x_3)\in [l_1, u_1]\times [l_2, u_2] \times[l_3, u_3]\big| \, x_1 x^{b_2}_2 = x_3\}, \] where \(l_1,l_2,l_3\in \mathbb{R}_{++} := \{x \in \mathbb{R}\big| \, x > 0\}\), \(u_1, u_2, u_3\in \mathbb{R}_{++}\), \( l_i\leq u_i\) for \(i = 1, 2, 3\), and \(b_1, b_2\geq 1\). Note that \(b_1\) and \(b_2\) are not restricted to be integers.