Let $M$ be a smooth manifold. Suppose $d$ is an intrinsic metric on $M$ which induces its topology.
Is it true that $(M,d)$ is locally convex? (i.e are there arbitrarily small convex neighbourhoods around every point?)
Is every length structure on a manifold locally convex?
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metric-spaces
manifolds
convex-analysis
smooth-manifolds
metric-geometry
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0What is an "intrinsic metric"? Also, convexity is usually a concept associated with certain paths, for instance line segments in Euclidean geometry. What paths are you using? – 2017-01-02
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0@LeeMosher Intrinsic metric is a metric which "induces itself" (every metric induces a *length structure* on the class of continuous paths which induces a new distance - by taking the infimum of the lengths of connecting paths). The originnal metric is called intrinsic if it coincides with the induced metric. (see ehttps://en.wikipedia.org/wiki/Intrinsic_metric). I use here continuous paths (some of them might be non-rectifiable of course), but I think nothing is supposed to change if we use only smooth paths. – 2017-01-02
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0Finally, note that in the classical Riemannian case, the metric spaces are always locally convex: http://mathoverflow.net/questions/252605/are-small-varepsilon-balls-convex-in-geodesic-metric-spaces. – 2017-01-02
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0I am still confused by how convexity is defined. I would guess that a subset is convex if each pair of points $p,q$ in the subset are the endpoints of some path in a specified collection of paths. But I do not know which collection of paths you wish to specify. Certainly you don't mean all continuous paths, because with respect to that collection every path connected subspace is convex. – 2017-01-02
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0@LeeMosher Roughly, a subset $U \subseteq X$ is convex if every two points in $U$ are connected by a *length-minimizing* path which is contained in $U$. – 2017-01-02
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0Okay, I think the question is clear now. – 2017-01-02
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0However, you might wish to strengthen your notion of metric to include the requirement that length minimizing paths exist. Otherwise, there might be strange counterexamples where the infimum of lengths of paths connecting two points exists but is not realized. – 2017-01-02
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0The Heisenberg group $H_3$ equipped with the Carnot-Caratheodory metric is a good candidate for a counter example. – 2017-01-03
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0It looks like you were right: http://mathoverflow.net/a/252676/46290. – 2017-01-03
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0Also http://www.math.unipd.it/~monti/PAPERS/JCA.pdf – 2017-01-04