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For and convex

For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.

For the preceding property of unions of non-decreasing sequences of convex sets, the restriction to nested sets is important: The union of two convex sets need not be convex.

For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is defined to be the smallest such distance.

For a set of scattered points in the plane, the diameter of the points is the same as the diameter of their convex hull.

For example the central convex pentagon in the center of a pentagram has density 2.

For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape formed by a rubber band stretched around X.

For each choice of coefficients, the resulting convex combination is a point in the convex hull, and the whole convex hull can be formed by choosing coefficients in all possible ways.

For points in two and three dimensions, output-sensitive algorithms are known that compute the convex hull in time O ( n log h ).

For dimensions d higher than 3, the time for computing the convex hull is, matching the worst-case output complexity of the problem.

* For a differentiable Lipschitz map ƒ: U → R m the inequality holds for the best Lipschitz constant of f, and it turns out to be an equality if the domain U is convex.

For n = 2 the theorem claims that a convex figure in the plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin.

For instance, if we abstract sets of couples ( x, y ) of real numbers by enclosing convex polyhedra, there is no optimal abstraction to the disc defined by x 2 + y 2 ≤ 1.

For a converging lens ( for example a convex lens ), the focal length is positive, and is the distance at which a beam of collimated light will be focused to a single spot.

For any convex, real-valued function φ such that

For a convex polygon ( such as a triangle ), a surface normal can be calculated as the vector cross product of two ( non-parallel ) edges of the polygon.

For instance, a ( strictly ) convex function on an open set has no more than one minimum.

For a twice-differentiable function f, if the second derivative, f ′′( x ), is positive ( or, if the acceleration is positive ), then the graph is convex ; if f ′′( x ) is negative, then the graph is concave.

For a real convex function, numbers x 1 , x 2 , ..., x n in its domain, and positive weights a i , Jensen's inequality can be stated as:

For example, when the space of functions is a Banach space, the functional derivative becomes known as the Fréchet derivative, while one uses the Gâteaux derivative on more general locally convex spaces.

For and two-dimensional

For a two-dimensional array, the element with indices i, j would have address B + c · i + d · j, where the coefficients c and d are the row and column address increments, respectively.

For example, a similar geometric system was published in 1987 by Kjell Gustafson, whose method represents a rhythm as a two-dimensional graph.

For convenience, pixels are normally arranged in a regular two-dimensional grid.

For example, Girih tiles in a medieval Islamic mosque in Isfahan, Iran, are arranged in a two-dimensional quasicrystalline pattern.

For example, the surface of the Earth is ( ideally ) a two-dimensional sphere, and latitude and longitude provide two-dimensional coordinates on it ( except at the poles and along the 180th meridian ).

For a two-dimensional situation with horizontal and vertical forces, the sum of the forces requirement is two equations: ΣH

For a simple visual explanation of a wormhole, consider spacetime visualized as a two-dimensional ( 2D ) surface.

For two-dimensional signals, e. g., images, the spectrum is reciprocal to f 2 .

For two-dimensional rotational motion, Newton's second law can be adapted to describe the relation between torque and angular acceleration:

For example, is a meromorphic function on the two-dimensional complex affine space.

For a two-dimensional surface, the Bianchi identities imply that the Riemann tensor can be expressed as

For example, atmospheric cyclones are rotational but their substantially two-dimensional shapes do not allow vortex generation and so are not turbulent.

For example, the ordinary generating function of a two-dimensional array a m, n ( where n and m are natural numbers ) is

For instance, a "< big > b </ big >" and a "< big > d </ big >" have a different shape, at least when they are constrained to move within a two-dimensional space like the page on which they are written.

For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

For example, a monatomic gas with a fixed number of particles is a simple case of a two-dimensional system ().

For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives.

For a two-dimensional flow the vorticity acts as a measure of the local rotation of fluid elements.

For a two-dimensional analysis a plane stress or a plane strain condition can be assumed.

For example, Kahane uses Cantor sets to construct a Besicovitch set of measure zero in the two-dimensional plane.

For such models, economists often use two-dimensional graphs instead of functions.

For example, a two-dimensional real torus has a SL ( 2, Z ) group of large diffeomorphisms by which the one-cycles of the torus are transformed into their integer linear combinations.

For the graph of a two-dimensional function, this corresponds to a point on the graph where the tangent plane is parallel to the xy plane.

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