Exact values of constants in the generalized triangle inequality for some \((1, q_2)\)-quasimetrics on canonical Carnot groups (Q276696)

Generalized triangle inequalities of the form \(d(u,v)\leq q_1d(u,w)+q_2d(w,v)\) are an important tool in studying graded stratified Lie algebras. The article studies the case when \(q_1=1\) and evaluates the constant \(q_2\) for the Carnot-Carathéodory distance using equivalent box quasimetric. The results are obtained in an explicit form for the following canonical Carnot groups: the Heisenberg group \(H^n_{\alpha}\) for \(n=1,2,\ldots\); the group \(H_{\alpha_1,\ldots, \alpha_n}\), and the Engel group \(E_{\alpha, \beta}\).