Metric Segments
Let be a metric space (which is not necessarily convex). A subset of is called a metric segment between two distinct points and in if there exists a closed interval on the real line and an isometry
such that and
It is clear that any point in such a metric segment except for the "endponts" and is between and As such, if a metric space admits metric segments between any two distinct points in the space, then it is a convex metric space.
The converse is not true, in general. The rational numbers form a convex metric space with the usual distance, yet there exists no segment connecting two rational numbers which is made up of rational numbers only. If however, is a convex metric space, and, in addition, it is complete, one can prove that for any two points in there exists a metric segment connecting them (which is not necessarily unique).
Read more about this topic: Convex Metric Space
Famous quotes containing the word segments:
“It is not, truly speaking, the labour that is divided; but the men: divided into mere segments of men—broken into small fragments and crumbs of life, so that all the little piece of intelligence that is left in a man is not enough to make a pin, or a nail, but exhausts itself in making the point of a pin or the head of a nail.”
—John Ruskin (1819–1900)