Seonglae ChoInterpolated points are contained in the set
In general, a set or function is convex if, for any two points within it, the line segment connecting them lies entirely within the set or below/on the function's graph. In the perspective of function, function is convex if the line segment between any two points on its graph lies below or on the graph
- When any two points are selected, all points on the line segment connecting these two points also belong to the function or set.
- When any two points are selected, if all points between these two points belong to the function or set, we say that this function or set is strictly convex.
Second Derivative positive is equivalent to convexity. In a same sense, the multivariate is convex iff Hessian Matrix’s all Eigenvalue is larger than 0.
Strictly Convex
There should not be flat regions. Iff Hessian Matrix’s all Eigenvalue > 0, there are no flat regions and strictly convex.
Properties
- The sum of any two convex functions is convex
- The maximum of any two convex functions is convex