Convex

Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.

A lens is biconvex (or double convex, or just convex) if both surfaces are convex.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.

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 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.

Non-Euclidean geometry The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set.

A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set.

A lens with one convex and one concave side is convex-concave or meniscus.

A set that is not convex is called a non-convex set.

As the cross product of a finite number of compact convex sets, is also compact and convex.

Computation of convex hulls main In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects.

Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape.

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

He also described an improved telescope—now known as the astronomical or Keplerian telescope —in which two convex lenses can produce higher magnification than Galileo's combination of convex and concave lenses.

However, they are special cases of a more general definition that is valid for any kind of n-dimensional convex or non-convex object, such as a hypercube or a set of scattered points.

If binding around a not fully convex, or square-edged object, arrange the knot so the overhand knot portion is stretched across a convex portion, or a corner, with the riding turn squarely on top of it.

These results show that Minkowski addition differs from the union operation of set theory ; indeed, the union of two convex sets need not be convex: The inclusion Conv(S) ∪ Conv(T) ⊆ Conv(S ∪ T) is generally strict.

This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces.

To use this method, every proposed district is circumscribed by the smallest possible convex polygon (similar to the concept of a convex hull ; think of stretching a rubberband around the outline of the district).