Box-Quasimetrics and Horizontal Joinability on Cartan Groups

On a Cartan group \({\mathbb{K}}\) equipped with a Carnot–Carathéodory metric d_cc, we find the exact value of a constant in the (1, q₂)-generalized triangle inequality for its Box-quasimetric. It is proved that any two points x, y ∈ \({\mathbb{K}}\) can be joined by a horizontal k-broken line \({L}_{x,y}^{k}\), k ≤ 6; moreover, the length of such a broken line \({L}_{x,y}^{k}\) does not exceed the quantity Cd_cc(x, y) for some constant C not depending on the choice of x, y ∈ \({\mathbb{K}}\). The value 6 here is nearly optimal.