Let $ (X,d)$ be a Cat(0) space of dimension 2. Following the notation of “Metric spaces of non-positive curvature” by Bridson and Haefliger, I define:
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a geodesic triangle $ \Delta\subset X$ consists of three points $ p,q,r\in X$ , its vertices, and a choice of three geodesic segments joining them, its sides.
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A subset $ C$ of $ X$ is said to be convex if every pair of points $ x,y\in C$ can be joined by a geodesic in $ X$ and the image of every such geodesic is contained in $ C$
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the (closed) convex hull of a subset $ A$ of a geodesic space $ X$ is the intersection of all (closed) convex subspaces of $ X$ containing $ A$ .
Since $ (X,d)$ is of dimension 2, it makes sense to talk about the interior of a geodesic triangle $ \Delta$ : it is the bounded subset of $ X$ delimited by $ \Delta$ . Is it true that the convex hull of every geodesic triangle $ \Delta$ always consists of the union of $ \Delta$ with its interior? If not, could you show a counterexample?
