TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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Finite Jensen inequality for convex functions on $R^{n}$

Statement

Let $f : C \to R$ be convex. For a positive finite family of points in $C$ and nonnegative weights summing to one, $f$ of their weighted Euclidean sum is at most the weighted sum of their $f$ -values. Explicitly, if $N \geq 1$ , $x_{1}, \dots, x_{N} \in C$ , $λ_{j} \geq 0$ , and $\sum_{j = 1}^{N} λ_{j} = 1$ , then

$f (\sum_{j = 1}^{N} λ_{j} x_{j}) \leq \sum_{j = 1}^{N} λ_{j} f (x_{j}) .$

Facts & Assumptions

Given: The data in the Statement, the finite-sum convention Finite sums and finite products, by recursion, its algebraic laws Laws of finite sums and finite products, and induction on the positive integer $N$ The principle of mathematical induction.

[F1]

The function $f : C \to R$ is convex when $f ((1 - t) x + t y) \leq (1 - t) f (x) + t f (y)$ for all $x, y \in C$ and $t \in [0, 1]$ (Convex and strictly convex functions on Euclidean convex sets).

Proof

technique · induction

1.1

For $N = 1$ , the only nonnegative weight summing to one is $λ_{1} = 1$ , so both sides equal $f (x_{1})$ .

F1algebrabase

1.2

Fix $N \geq 1$ and assume the inequality for every weighted family of $N$ points.

2.1

For $N + 1$ points, if $λ_{N + 1} = 1$ , then all earlier nonnegative weights vanish and the conclusion is immediate. Otherwise $s = 1 - λ_{N + 1} > 0$ , and the normalized weights $μ_{j} = λ_{j} / s$ for $j \leq N$ are nonnegative and sum to one.

step 1.1step 1.2algebra

3.1

Apply the induction hypothesis to $z = \sum_{j = 1}^{N} μ_{j} x_{j} \in C$ , then apply [F1] to $z, x_{N + 1}$ with weights $s, λ_{N + 1}$ . The resulting inequality is exactly the $(N + 1)$ -point formula, completing the induction.

step 1.2step 2.1F1algebradischarge-induction∎

Depends on

Used by

Dependency tree · two levels

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Sources

S. Boyd and L. Vandenberghe, Convex Optimization, §3.1.8 (standard reference, not scraped)

Finite Jensen inequality for convex functions on RnRn