Definition:Diameter of Bounded Subset of Connected Riemannian Manifold

Definition

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.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances