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Let $X$ be an inner metric space with curvature bounded from below by $k$ in the sense of Toponogov. $\Sigma_p$ be the space of directions at point $p$. In the note by Plaut "Metric spaces of curvature bounded from below", the author mentioned thesis of Stephanie Gloor (1998, Zurich), which contains an example of an inner metric space with curvature $\ge k$ such that the space of directions at some point is not an inner metric space.

Does any one know this example?

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up vote 3 down vote accepted

I'm not familiar with that reference but a standard example for this is by Stephanie Halbeisen "On tangent cones of Alexandrov spaces with curvature bounded below".

The example is necessarily infinite dimensional as it's well known that this can not happen in finite dimensions. Also, it's worth noting that there is no canonical definition of a space of directions for infinite dimensional Alexandrov spaces. The definition that Halbeisen uses is a metric completion of equivalence classes of geodesic segments starting at $p$. But there are other natural definitions possible (say, by looking at ultralimit of pointed blow ups of $X$ at $p$). All these definitions agree for finite dimensional Alexandrov spaces but not for infinite dimensional ones.

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