Convex Metric Space
In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.
Formally, consider a metric space (X, d) and let x and y be two points in X. A point z in X is said to be between x and y if all three points are distinct, and
that is, the triangle inequality becomes an equality. A convex metric space is a metric space (X, d) such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y.
Metric convexity:
- does not imply convexity in the usual sense for subsets of Euclidean space (see the example of the rational numbers)
- nor does it imply path-connectedness (see the example of the rational numbers)
- nor does it imply geodesic convexity for Riemannian manifolds (consider, for example, the Euclidean plane with a closed disc removed).
Read more about Convex Metric Space: Examples, Metric Segments, Convex Metric Spaces and Convex Sets
Famous quotes containing the word space:
“Sir Walter Raleigh might well be studied, if only for the excellence of his style, for he is remarkable in the midst of so many masters. There is a natural emphasis in his style, like a man’s tread, and a breathing space between the sentences, which the best of modern writing does not furnish. His chapters are like English parks, or say rather like a Western forest, where the larger growth keeps down the underwood, and one may ride on horseback through the openings.”
—Henry David Thoreau (1817–1862)