Concepts inUsing convex hulls to represent classifier conditions
Convex hull
In mathematics, the convex hull or convex envelope for a set X of points in the Euclidean plane or Euclidean space is the minimal convex set containing X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape formed by a rubber band stretched around X. Formally, the convex hull may be defined as the intersection of all convex sets containing X, the intersection of all halfspaces containing X, or the set of all convex combinations of points in X.
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Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. The notion can be generalized to other spaces as described below.
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Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ¿ x ¿ 1 is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers, the set of all negative real numbers, and the empty set.
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Convex function
In mathematics, a real-valued function defined on an interval is called convex (or convex downward or concave upward) if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. More generally, this definition of convex functions makes sense for functions defined on a convex subset of any vector space.
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Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term ¿Euclidean¿ distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria.
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Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces.
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Candidate solution
In optimization (a branch of mathematics) and search algorithms (a topic in computer science), a candidate solution is a member of a set of possible solutions to a given problem. A candidate solution does not have to be a likely or reasonable solution to the problem ¿ it is simply in the set that satisfies all constraints. The space of all candidate solutions is called the feasible region, feasible set, search space, or solution space.
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