Definition:Diameter of Bounded Subset of Connected Riemannian Manifold

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Let $\struct {M, g}$ be a connected Riemannian manifold.

Let $p,q \in M$ be points.

Let $d_g$ be the Riemannian distance.

Let $A \subseteq M$ be a bounded subset of $M$.

Then the diameter of $A$, denoted by $\map \diam A$ is defined as:

$\map \diam A := \map \sup {\map {d_g} {p, q} : p, q \in A}$

where $\sup$ denotes the supremum.