Jensen's Inequality Jensen's inequality is a fundamental inequality in convex geometry that establishes a relationship between the values of a function on a...
Jensen's Inequality
Jensen's inequality is a fundamental inequality in convex geometry that establishes a relationship between the values of a function on a closed convex set and the value of the function at the set's center.
Theorem: If f is a function defined on a closed convex set S, and x is an element of S, then the following inequality holds:
where p is the Hausdorff dimension of the set S.
Proof: Jensen's inequality can be proven using several methods, including the following:
Geometric interpretation: If we consider the function f(x) as the area of the region bounded by the graph of the function, then Jensen's inequality can be interpreted as the statement that the area of the smaller region containing the point x must be less than or equal to the area of the larger region.
Geometric interpretation: Jensen's inequality can also be proven by observing that the left-hand side of the inequality can be interpreted as the distance from x to the center of the set S. The right-hand side of the inequality then gives the maximum distance from x to any point in the set.
Functional interpretation: Jensen's inequality can be derived from the fact that the function f(x) is a convex function. A convex function always takes values closer to its minimum than it does to its maximum. Therefore, if we take the minimum value of f(x) on the boundary of S, we must have f(x) \le \left\lVert x - \frac{x}{|x|} \right\rVert^p.
Examples:
If S is the unit disk, and p=2, then Jensen's inequality becomes (0 \le x \le 1), which is a classical result in geometry.
If S is the half-space ((-1, 1)), and p=1, then Jensen's inequality becomes (0 \le x \le 1), which is also a classical result in geometry