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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 .
In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any sets of points. [ 1 ] [ 2 ] Formal Definition
Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function. [11] A strongly convex function is also strictly convex, but not vice versa. If a one-dimensional function f {\displaystyle f} is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:
Carathéodory's theorem – Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P; Helly's theorem – Theorem about the intersections of d-dimensional convex sets; Krein–Milman theorem – On when a space equals the closed convex hull of its extreme points; List of convexity topics
In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
Convex analysis - the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization. Convex combination - a linear combination of points where all coefficients are non-negative and sum to 1. All convex combinations are within the convex hull of the given points.
In mathematics, variational analysis is the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. [1] This includes the more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.
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.