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The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n -ball , the n -ball of radius 1 .
The volume of the unit ball in Euclidean -space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit n {\\displaystyle n} -ball, which we denote V n , {\\displaystyle V_{n},} can be expressed by making use of the gamma function .
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...
A unit -sphere is the unit circle in the Euclidean plane, and its interior is the unit disk ( -ball). S 1 = 2 π , V 2 = π . {\displaystyle S_{1}=2\pi ,\quad V_{2}=\pi .} The interior of a 2-sphere in three-dimensional space is the unit 3 {\displaystyle 3} -ball.
There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1): . If A ⊆ S n−1 is any Borel set and B⊆ S n−1 is a ρ n-ball with the same σ n-measure as A, then, for any r > 0,
The interior of a 3-sphere is a 4-ball. ... The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere ... The 3-dimensional surface volume of a 3 ...
For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since V = π / 6 d 3, where d is the diameter of the sphere and also the length of a side of the cube and π / 6 ≈ 0.5236.
The amount of empty space is measured in the packing density, which is defined as the ratio of the volume of the spheres to the volume of the total convex hull. The higher the packing density, the less empty space there is in the packing and thus the smaller the volume of the hull (in comparison to other packings with the same number and size ...