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The nth root of the content of the set sum of two sets in n-dimensional Euclidean space is greater than or equal to the sum of the nth roots of the contents of the individual ...

The distance between two points is the length of the path connecting them. In the plane, the distance between points (x_1,y_1) and (x_2,y_2) is given by the Pythagorean ...

If M^n is a finite simplicial complex of dimension n>=5 that has the homotopy type of the sphere S^n and is locally piecewise linearly homeomorphic to the Euclidean space ...

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to ...

The metric of Felix Klein's model for hyperbolic geometry, g_(11) = (a^2(1-x_2^2))/((1-x_1^2-x_2^2)^2) (1) g_(12) = (a^2x_1x_2)/((1-x_1^2-x_2^2)^2) (2) g_(22) = ...

The metric ds^2=(dx^2+dy^2)/((1-x^2-y^2)^2) for the Poincaré hyperbolic disk, which is a model for hyperbolic geometry. The hyperbolic metric is invariant under conformal ...

In elliptic n-space, the pole of an (n-1)-flat is a point located at an arc length of pi/2 radians away from each point of the (n-1)-flat.

A method for finding solutions u and v to a linear congruence au+bv=d by constructing a matrix formed by adjoining a vector containing a and b with a unit matrix, M=[a 1 0; b ...

The dimension e(G), also called the Euclidean dimension (e.g., Buckley and Harary 1988) of a graph, is the smallest dimension n of Euclidean n-space in which G can be ...

The generalized diameter is the greatest distance between any two points on the boundary of a closed figure. The diameter of a subset E of a Euclidean space R^n is therefore ...