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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 geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset.
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.
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. Convex and Concave - a print by Escher in which many of the structure's features can be seen as both convex shapes and concave impressions.
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.
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [ 3 ] [ 4 ] [ 5 ] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph ∪ {\displaystyle \cup } .
For example, a solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non‑convex. Trivially, the empty set is convex. More formally, a set Q is convex if, for all points v 0 and v 1 in Q and for every real number λ in the unit interval [0,1], the point (1 − λ) v 0 + λv 1. is a member of Q.
It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under additions. More succinctly, a set C {\displaystyle C} is a convex cone if and only if α C = C {\displaystyle \alpha C=C} and C + C = C {\displaystyle C+C=C} , for any positive scalar α ...