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I am interested in subsets $A \subset M$ of a connected, complete Riemannian manifold $(M,g)$ with the following property: for every $p,q \in A$, at least one minimising geodesic from $p$ to $q$ in $M$ is contained in $A$. Is there a name for this property? It is more general than convexity and weak convexity as defined in Chavel's Riemannian Geometry: A Modern Introduction, since it includes e.g. `bands' on a cylinder.

1Rock
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    This is the same as definition (A2) in this question: https://math.stackexchange.com/questions/3198530/notions-of-convexity – 1Rock Apr 30 '20 at 02:57
  • Whitehead, 1931 ('Convex regions in the geometry of paths') refers to regions whose points are joined by unique paths as simple convex regions. Unfortunately, I think there's been a nomenclature change over the past 80 years, and now 'convex' means what Whitehead called 'simple convex'. – 1Rock Apr 30 '20 at 07:23

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