Affine hull

Affine hull

Nonlinear Programming

Published in Albert G. Holzman, Mathematical Programming, 2020

Consider a subset B of Rn with the following property. For every two points x1 ∈ B, x2 ∈ B and for every value of the real number α, also αx1 + (1 - α)x2 ∈ B. A set satisfying this property is called an affine set. Single points, lines, and hyperplanes are examples of affine sets in Rn. Given a convex set C⊂Rn , the intersection of all affine sets containing C is called the affine hull of C. The relative interior of C, denoted by ri C, is defined as the interior of C, viewed as a subset of its affine hull.

On Some Important Inequalities

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Published in Hemen Dutta, Topics in Contemporary Mathematical Analysis and Applications, 2020

Zlatko Pavić

Since the differences a1 − a0,…,an − a0 are linearly independent, the affine combination in formula (7.10) is unique for each x ∈ ℝn. The affine hull of the set of simplex vertices covers the space ℝn, i.e., 𝔸a0…an = aff{a0,…,an} = ℝn.

Moderate deviations for stochastic variational inequalities

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Journal Information

Published in Optimization, 2023

Mingjie Gao, Ka-Fai Cedric Yiu

A subset K of is called a cone if whenever and μ is a positive real number. For a set , or denotes its interior, and is the smallest cone that contains it. A subset C of is said to be affine if for every and real number λ. The affine hull of a convex set C is defined to be the intersection of all the affine sets which contains C. The related interior of C is defined as the interior which results when C is regarded as a subset of .

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