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An unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A sphere or ball with unit radius and center at the origin of the space is called the unit sphere or the unit ball.
Volumes of balls in dimensions 0 through 25; unit ball in red. In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space.
In particular, a ball (open or closed) always includes p itself, since the definition requires r > 0. A unit ball (open or closed) is a ball of radius 1. A ball in a general metric space need not be round. For example, a ball in real coordinate space under the Chebyshev distance is a hypercube, and a ball under the taxicab distance is a cross ...
An alternative given by Marsaglia is to uniformly randomly select a point = (,, …,) in the unit n-cube by sampling each independently from the uniform distribution over (,) , computing as above, and rejecting the point and resampling if (i.e., if the point is not in the -ball), and when a point in ...
It was introduced by David Hilbert as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the n-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces.
Although the -unit ball around the origin in this metric is "concave", the topology defined on by the metric is the usual vector space topology of , hence is a locally convex topological vector space.
In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at ...
The above formula is for the xy plane passing through the center of mass, which coincides with the geometric center of the cylinder. If the xy plane is at the base of the cylinder, i.e. offset by d = h 2 , {\displaystyle d={\frac {h}{2}},} then by the parallel axis theorem the following formula applies: