Convex Sets
A convex set is a collection of points where, if you take any two points within the set and draw a straight line between them, the entire line will also lie within the set. Think of it like a rubber band stretched around objects: if the band encloses a shape, all points inside it form a convex set. Conversely, if you have a shape where a line drawn between two points goes outside the shape, that set is not convex. Examples of convex sets include circles, triangles, and squares, whereas a crescent moon shape is not convex.
Additional Insights
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A convex set is a collection of points where, for any two points in the set, the entire line segment connecting them also lies within the set. Imagine a rubber band stretched around a group of points—if the band encloses them, the area inside is convex. Common examples include a circle, triangle, or square. If a shape allows a line segment between points to extend outside its boundaries, it is not convex. Convexity is important in various fields, including geometry, economics, and optimization, as it simplifies many problems and models.