In this wikipedia page https://en.wikipedia.org/wiki/Sectional_curvature I've found a Theorem by Toponogov which characterizes Riemannian spaces of non positive curvature:
Thm 1 If a complete Riemannian manifold is of non positive curvature, then for all sufficiently small geodesic triangles $xyz$, given $m$ the midpoint of the side $xy$ it is verified
$d(z,m)^2\le \frac 1 2 d(z,x)^2+\frac 1 2 d(z,y)^2-\frac 1 4 d(x,y)^2$
where $d$ is the distance induced by the Riemannian metric.
A few lines below it is stated that a simple consequence of this theorem is the following:
Thm 2 A complete simply connected Riemannian manifold has non-positive sectional curvature if and only if the function $f_p(x):=d(p,x)^2$ is 1-convex (which I think should mean that, for every geodesic $\gamma(t)$, the function $f_p(\gamma(t))-t^2$ is convex).
I have two questions:
1) Is the inverse implication of Thm 1 true (if the inequality is verified for every triangle then the space has non positive curvature)?
2) Do you know how to derive Thm 2 from Thm 1 (in the wikipedia page it is written that it's a simple consequence..)?
Also can you give me a good reference on these Toponogov's theorems?