Let $\kappa > 0$. I have a space $(X,d)$ and take 4 points $w,x,y,z \in X$. I then choose comparison points in the model space $(M\_\kappa^2,\bar{d})$ as follows:
Take the comparison triangle $\Delta xyz$ with same sidelengths (and points $\bar{x}, \bar{y}, \bar{z}$). Furthermore I take the comparison triangles
$\Delta xyw$, $\Delta yzw$, $\Delta xzw$ where I can choose the comparison points such that we have the points $\bar{x}, \bar{y}, \bar{z}$ from before and the possibly different points $\bar{w\_1}, \bar{w\_2}, \bar{w\_3}$ corresponding to each of the triangles. If it now holds that 


1. $d(w,z) < \bar{d}(\bar{w\_1}, \bar{z})$,
2. $d(w,x) < \bar{d}(\bar{w\_2}, \bar{x})$ and
3. $d(w,y) < \bar{d}(\bar{w\_3}, \bar{y})$


Does this prevent the space $X$ from being CBB($\kappa$)?



Inequalities on 4 points in metric spaces with curvature bounded below (CBB)
No. Take a $X = \{x, y, z, w\} $ to be such that all distances are equal to $ 1$. Take a very small $k$. Consider a $3$-dimensional sphere with the curvature $k$ and taken with its intrinsic metrics. (So this sphere is a giant. ) We can embedd $ X$ isometrically into this sphere. (Just draw a rigth tetrahedron in some tangent space with the center of tetrahedron at $0$. And then take the exponential image.) 


On the other hand the comparison space $M^2\_k$ also is a giant sphere but a two dimensional one. So if we draw the comparison picture such that all comparison points $x', y', z', w'\_1, w'\_2, w'\_3$ are distinct then the required inequalities will be satisfied. 


