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For example, a real-valued Lebesgue measurable function that is midpoint-convex is convex: this is a theorem of Sierpiński. [8] In particular, a continuous function that is midpoint convex will be convex.
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions.
The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum. Convex polygon - a 2-dimensional polygon whose interior is a convex set in the Euclidean plane. Convex polytope - an n-dimensional polytope which is also a convex set in the Euclidean n-dimensional ...
Equivalently, a convex set or a convex region is a set that intersects every line in a line segment, single point, or the empty set. [1] [2] For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set in the plane is always a convex curve.
An example of a convex polygon: a regular pentagon. In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). [1]
For example, a solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non‑convex. Trivially, the empty set is convex. More formally, a set Q is convex if, for all points v 0 and v 1 in Q and for every real number λ in the unit interval [0,1], the point (1 − λ) v 0 + λv 1. is a member of Q.
Any convex set in a Euclidean space is a convex metric space with the induced Euclidean norm. For closed sets the converse is also true: if a closed subset of a Euclidean space together with the induced distance is a convex metric space, then it is a convex set (this is a particular case of a more general statement to be discussed below).
An affine convex cone is the set resulting from applying an affine transformation to a convex cone. [8] A common example is translating a convex cone by a point p: p + C. Technically, such transformations can produce non-cones. For example, unless p = 0, p + C is not a linear cone. However, it is still called an affine convex cone.