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The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets = = ( ()). The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice .
Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap .
The convex hull of the red set is the blue and red convex set.. In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it.
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.
A closed convex subset of a topological vector space is called strictly convex if every one of its (topological) boundary points is an extreme point. [6] The unit ball of any Hilbert space is a strictly convex set. [6]
Convex set – In geometry, set whose intersection with every line is a single line segment; Locally convex topological vector space – A vector space with a topology defined by convex open sets; Radial set; Star domain – Property of point sets in Euclidean spaces; Symmetric set – Property of group subsets (mathematics)
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.
A set is relatively open iff it is equal to its relative interior. Note that when aff ( S ) {\displaystyle \operatorname {aff} (S)} is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed.