# Inequalities on 4 points in metric spaces with curvature bounded below (CBB)

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($$\kappa$$)?

• Are you aware of the "4-point condition" for CAT(k) spaces, i.e., for CBA spaces, cf. Def. II.1.10., p. 164 in Metric spaces of non-positive curvature, Bridson-Haefliger? This looks similar and might give you some ideas... – Clemens Sämann Jun 21 '19 at 15:45
• @ClemensSämann Yes but may need to check this again. Did not think I can use it here at first. – Loreno Heer Jun 21 '19 at 15:48
• @ClemensSämann As far as I understand the 4-point CAT condition would only work to show that the space can not be CAT. The 4-point condition for CBB spaces looks slightly different (7.1.3 in Alexandrov geometry: preliminary version no. 1, Alexander, Kapovitch, Petrunin) in the sense of which points are compared. – Loreno Heer Jun 21 '19 at 18:43

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.