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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($\kappa$)?

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  • $\begingroup$ 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... $\endgroup$ – Clemens Sämann Jun 21 '19 at 15:45
  • $\begingroup$ @ClemensSämann Yes but may need to check this again. Did not think I can use it here at first. $\endgroup$ – Loreno Heer Jun 21 '19 at 15:48
  • $\begingroup$ @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. $\endgroup$ – Loreno Heer Jun 21 '19 at 18:43
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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.

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