Important inequalities: the convex inequality

Ask the tutor about Important inequalities: the convex inequality Open the graphing workbench

Square the average of two real numbers, or average the two squares. Which number is larger? Equal weights and the square are a narrow case. Several points, unequal weights, and a density on a region still need a function that bends the same way.

Two points on the square

The board freezes three extra assumptions: the function is $t \mapsto t^{2}$ , there are two real sample points, and one weight runs from $0$ to $1$ .

Drag a, b, or w. The curve is t squared. a and b are the sample points, and w runs from 0 to 1, so the other weight is what remains. The point on the curve is the square of the weighted average. The point on the chord is the weighted average of the two squares. The segment between them is the gap. Bring a and b together, or push w to either end, and the gap closes.

The point on the curve stays on or below the chord. Bring the sample points together, or push the weight to either end, and the gap closes.

Let $m = 1, 2, \dots$ . Let $F : R^{N} \to R$ be convex. For points $x_{k} \in R^{N}$ and nonnegative real weights $λ_{k}$ with $\sum_{k = 1}^{m} λ_{k} = 1$ ,

$F (\sum_{k = 1}^{m} λ_{k} x_{k}) \leq \sum_{k = 1}^{m} λ_{k} F (x_{k}) .$

The board used $N = 1$ , $m = 2$ , and one convex function, the square. The weights there are $1 - w$ and $w$ .

A density in place of the weights

The same comparison on a region uses a nonnegative density. Here $F$ reads a real number, the value of $g$ , and no longer reads a point of $R^{N}$ . The point of $R^{N}$ is where $g$ and the density are evaluated.

$F (\frac{\int _{G} p ( x ) g ( x ) d x}{\int _{G} p ( x ) d x}) \leq \frac{\int _{G} p ( x ) F ( g ( x )) d x}{\int _{G} p ( x ) d x} .$

This comparison on a region is (0.22).

Three hypotheses. $F : R \to R$ is convex. The density $p : G \to R$ is nonnegative and integrable on an open set $G \subset R^{N}$ , and $\int_{G} p d x > 0$ . The function $g : G \to R$ is such that every integral in that comparison exists.

The constant density $p (x) \equiv 1$ meets the hypothesis on $p$ wherever that integral is positive.

Drag c or s. The density is frozen as the constant 1, and the interval is [c, c+s]. g is a straight line, so its values run from c to c+s and the picture is that range. The curve is the square. The shaded region is the squares along that range. The horizontal bar is their average height. The point on the curve is the square of the average. Narrow s and the gap shrinks.

The point on the square stays below the average height of the square. Narrow the interval and the gap shrinks.

A convex function of a convex combination stays at most that same combination of the values. On a region the combination is the density, and $F$ reads only the real value $g (x)$ . The constant density $1$ is allowed. Changing the exponent on one fixed vector is a different comparison, Jensen's comparison of $p$ -norms.