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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.
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 .
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.
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions.
An affine convex cone is the set resulting from applying an affine transformation to a convex cone. [10] A common example is translating a convex cone by a point p : p + C . Technically, such transformations can produce non-cones.
In psychology, a set is a group of expectations that shape experience by making people especially sensitive to specific kinds of information. A perceptual set, also called perceptual expectancy, is a predisposition to perceive things in a certain way. [1] Perceptual sets occur in all the different senses. [2]
This definition is equivalent to the definition of convex curves from support lines. Every convex curve, defined as a curve with a support line through each point, is a subset of the boundary of its own convex hull. Every connected subset of the boundary of a convex set has a support line through each of its points. [8] [9] [19]