A set \(C\) is called a convex set if it contains all the [Convex Combination] of the points in that set. Given a set \(C\), we can make it convex by taking the [Convex Hull] of the set.
Examples of Convex Sets:
- Empty set \(\{\emptyset \}\)
- Singleton (A set containing a single point)
- The whole \(R^n\) space.
- [Hyperplane]
- [Halfspace]
- [Ellipsoid]
Operations that preserve convexity:
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Intersection: If two sets \(S_1\) and \(S_2\) are convex, then \(S_1 \cap S_2\) is convex.
Proof:
Let \(x_1, x_2 \in S_1 \cap S_2\). This means that \(x_1\) and \(x_2 \in S_1\) and \(x_1\) and \(x_2 \in S_2\). Therefore \(\theta x_1 + (1-\theta) x_2 \in S_1\) since \(S_1\) is convex. Similarly \(\theta x_1 + (1-\theta) x_2 \in S_2\) since \(S_2\) is convex. Therefore we can conclude that \(\theta x_1 + (1-\theta) x_2 \in S_1\cap S_2\). Therefore we can say \(S_1 \cap S_2\) is convex.
Vector subspaces, [Affine Set]s and [Convex Cone]s are also closed under arbitrary number of intersections. An exaple of this is a [Polyhedron] .
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[Affine functions] : If \(S \subseteq R^n\) is a convex set and \(f:R^n \rightarrow R^m\) is an affine function then the image of \(S\) under \(f\) is \(f(S) = \{ f(x) \mid x \in S \}\). Similarly the inverse image of \(S\) under \(f\) such that \(f^{-1}(S) = \{ x \mid f(x) \in S\}\) is also convex.
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Linear fractional and Perspective transform