I have seen a new type of distance called Carnot-Caretheodory distance defined in page 4 of the following article link : http://imar.ro/journals/Mathematical_Reports/Pdfs/2017/3/3.pdf
I am trying to construct example of such distance and have a feeling that the Euclidean distance $d(x,y)=|x-y|$ in $\mathbb{R}^N$ is a Carnot-Caretheodory distance. But unable to prove.
Please help me with this one. Thank you very much.