Search results
Results From The WOW.Com Content Network
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.
A convex set can have more than one supporting hyperplane at a given point on its boundary. This theorem states that if S {\displaystyle S} is a convex set in the topological vector space X = R n , {\displaystyle X=\mathbb {R} ^{n},} and x 0 {\displaystyle x_{0}} is a point on the boundary of S , {\displaystyle S,} then there exists a ...
In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x and y meets ∂B only at ...
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 .
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.
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]
By definition, a convex combination of an indexed subset {v 0, v 1, . . . , v D} of a vector space is any weighted average λ 0 v 0 + λ 1 v 1 + . . . + λ D v D, for some indexed set of non‑negative real numbers {λ d} satisfying the equation λ 0 + λ 1 + . . . + λ D = 1. The definition of a convex set implies that the intersection of two ...