Convex set

A convex set

A non-convex set

In Euclidean space, a region is a convex set if the following is true. For any two points inside the region, a straight line segment can be drawn. If every point on that segment is inside the region, then the region is convex.

The point is that a convex curve forms the boundary of a convex set. So, any shape which is concave, or has a hollow, cannot be a convex set.

Convex Set Media

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  A function is convex if and only if its epigraph, the region (in green) above its graph (in blue), is a convex set.

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  Blaschke-Santaló (r, D, R) diagram for planar convex bodies. \mathbb{L} denotes the line segment, \mathbb{I}_{\frac{\pi}{3}} the equilateral triangle, \mathbb{RT} the Reuleaux triangle and \mathbb{B}_2 the unit circle.

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  Minkowski addition of sets. The sum of the squares Q1=[0,1]2 and Q2=[1,2]2 is the square Q1+Q2=[1,3]2.

Related pages

• Convex function