Curvature bounded below: a definition a la Berg-Nikolaev (Q600832)

The main result of the paper states that a complete metric space \(M\) with intrinsic metric \(\rho \) is an Aleksandrov space of curvature \(\geq 0\) if and only if, for every quadruple of points \(\left\{ p,x,y,z\right\} \subseteq M\), the following inequality holds: \[ \rho ^{2}\left( p,x\right) +\rho ^{2}\left( p,y\right) +\rho ^{2}\left( p,z\right) \geq \frac{\rho ^{2}\left( x,y\right) +\rho ^{2}\left( y,z\right) +\rho ^{2}\left( z,x\right) }{3}. \]