* Convex conjugate, the (" dual ") lower-semicontinuous convex function resulting from the Legendre – Fenchel transformation of a " primal " function

The Bohr – Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis.

Those recommendations aside, constrictor knots do function best on fully convex objects.

If the matrix is positive semidefinite matrix, then is a convex function: In this case the quadratic program has a global minimizer if there exists some feasible vector ( satisfying the constraints ) and if is bounded below on the feasible region.

It turns out that the matrix M is positive definite if and only if it is symmetric and its quadratic form is a strictly convex function.

This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if M is positive definite.

Let F ( x ) be an upper semi-continuous function of x, and that F ( x ) is a closed, convex set for all x.

Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima, where a local minimum x < sup >*</ sup > is defined as a point for which there exists some δ > 0 so that for all x such that

As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.

Many boosting algorithms fit into the AnyBoost framework, which shows that boosting performs gradient descent in function space using a convex cost function.

For any convex, real-valued function φ such that

* The characteristic function in convex analysis:

A piecewise linear function in two dimensions ( top ) and the convex polytopes on which it is linear ( bottom ).

The rate – distortion function of any source is known to obey several fundamental properties, the most important ones being that it is a continuous, monotonically decreasing convex ( U ) function and thus the shape for the function in the examples is typical ( even measured rate – distortion functions in real life tend to have very similar forms ).

The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in ( d + 1 )- dimensional space, by giving each point p an extra coordinate equal to | p |< sup > 2 </ sup >, taking the bottom side of the convex hull, and mapping back to d-dimensional space by deleting the last coordinate.

This Stella octangula | 24 equilateral triangle example is not a Johnson solid because it is not convex.

File: Johannes Kepler 1610. jpg | Johannes Kepler ( 1571 — 1630 ): used the accurate observations of Tycho Brahe to formulate three fundamental laws of planetary motion, described elliptical motion of planets around the sun, developed early telescopes, invented the convex eyepiece, discovered a means of determining the magnifying power of lenses.

File: NAMA Mycènes bouclier 1. jpg | Wall painting depicting a Mycenaean Greek " figure eight " shield with a suspension strap at the middle, 15th century BC, National Archaeological Museum, Athens-The faces of figure eight shields were quite convex.

A function ( in black ) is convex if and only if the region above its Graph of a function | graph ( in green ) is a convex set.

The 120-cell ( or hecatonicosachoron ) is a convex regular 4-polytope consisting of 120 Dodecahedron | dodecahedral Cell ( geometry ) | cells

For any f in L < sup > 1 </ sup >( G ), the distance between 0 and the closed convex hull in L < sup > 1 </ sup >( G ) of the left translates λ ( g ) f equals | ∫ f |.

An example of a convex polygon: a regular polygon | regular pentagon

Image: Cross-Pattee-Alisee. svg | With the ends of the arms convex and curved ; sometimes called " Alisee " ( French croix pattée alésée arrondie )

Total Demand ( economics ) | Demand for this seafood is a convex function of household income, making it a superior good.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices.

The first assumption is that in the case where the background is opaque ( i. e. ), the over operator represents the convex combination of and:

We know that is opaque and thus follows that is opaque, so in the above equation, each operator can be written as a convex combination:

Alpha blending is a convex combination of two colors allowing for transparency effects in computer graphics.

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

A burning glass or burning lens is a large convex lens that can concentrate the sun's rays onto a small area, heating up the area and thus resulting in ignition of the exposed surface.

It is played on a pitch which may be flat ( for " flat-green bowls ") or convex or uneven ( for " crown-green bowls ").

As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively.

* Convex cone, a subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C

It has a convex, elliptical surface and is divided into anterior and posterior lips.

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object.

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.

A set C in S is said to be convex if, for all x and y in C and all t in the interval, the point

A set C is called absolutely convex if it is convex and balanced.

* In the plane ( d = 2 ), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face ( see Euler characteristic ).

In fact, a bounded linear functional on C < sub > c </ sub >( X ) need not remain so if the locally convex topology on C < sub > c </ sub >( X ) is replaced by the supremum norm, the norm of C < sub > 0 </ sub >( X ).

A set of points is defined to be convex if it contains the line segments connecting each pair of its points.

In fact, according to Carathéodory's theorem, if X is a subset of an N-dimensional vector space, convex combinations of at most N + 1 points are sufficient in the definition above.

If the convex hull of X is a closed set ( as happens, for instance, if X is a finite set or more generally a compact set ), then it is the intersection of all closed half-spaces containing X.

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

Suppose that L is a lattice of determinant d ( L ) in the n-dimensional real vector space R < sup > n </ sup > and S is a convex subset of R < sup > n </ sup > that is symmetric with respect to the origin, meaning that if x is in S then − x is also in S.

The area bound is sharp: if S is the interior of the square with vertices (± 1, ± 1 ) then S is symmetric and convex, has area 4, but the only lattice point it contains is the origin.