In functional analysis, the girth of a Banach space is the infimum of lengths of centrally symmetric simple closed curves in the unit sphere of the space.
[1][2] Every finite-dimensional Banach space has a pair of opposite points on the unit sphere that achieves the minimum distance, and a centrally symmetric simple closed curve that achieves the minimum length.
[1] The girth is always at least four, because the shortest path on the unit sphere between two opposite points cannot be shorter than the length-two line segment connecting them through the origin of the space.
There exist flat Banach spaces of infinite dimension in which the girth is achieved by a minimum-length curve; an example is the space C[0,1] of continuous functions from the unit interval to the real numbers, with the sup norm.
The unit sphere of such a space has the counterintuitive property that certain pairs of opposite points have the same distance within the sphere that they do in the whole space.