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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.
In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.
According to the Krein–Milman theorem, every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space) is the convex hull of its extreme points. [15] However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex ...
A conical combination is a linear combination with nonnegative coefficients. When a point is to be used as the reference origin for defining displacement vectors, then is a convex combination of points ,, …, if and only if the zero displacement is a non-trivial conical combination of their respective displacement vectors relative to .
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. By mathematical induction, a set Q is convex if and only if every convex combination of members of Q also belongs to Q.
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three.
Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's theorem , the body's perimeter is exactly π times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest ...