Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Analysis

Jensen's Inequality

For finitely many points in a convex domain, nonnegative weights that add to one produce a weighted barycenter; a convex function's value there is no greater than the same weighted average of its values at the original points.

Exact theorem

Exact Mathlib statement

theorem ConvexOn.map_sum_le {𝕜 E β ι : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [AddCommGroup β] [PartialOrder β] [IsOrderedAddMonoid β] [Module 𝕜 E] [Module 𝕜 β] [IsStrictOrderedModule 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i)

The theorem at a glance

Jensen's inequality at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

A convex function evaluated at a normalized finite weighted sum does not exceed the corresponding weighted sum of its values. Explanatory diagram.
Detailed visual description

The poster leads with the finite Jensen formula. Beneath it, several cobalt source-value markers on a convex bowl send antique-gold weighted threads to one emerald barycenter value; a separate vertical bracket keeps that value at or below the weighted-value level. Four hypothesis cards retain convexity on s, nonnegative weights, total weight one, and membership of every indexed point in s. The footer excludes integral, strict, equality-case, and concave variants.

Statement structure

From hypotheses to conclusion

Statement map for Jensen's InequalityConvexity bounds the value at a normalized finite weighted sum by the corresponding weighted sum of the values. The pinned upstream declaration is ConvexOn.map_sum_le. The exact checked statement is theorem ConvexOn.map_sum_le {𝕜 E β ι : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [AddCommGroup β] [PartialOrder β] [IsOrderedAddMonoid β] [Module 𝕜 E] [Module 𝕜 β] [IsStrictOrderedModule 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i).Mathematical readingConvexity bounds thevalue at a normalizedfinite weighted sum bythe correspondingweighted sum of thevalues.Pinned declarationmathlib ·ConvexOn.map_sum_leExact checked formtheoremConvexOn.map_sum_le {𝕜 Eβ ι : Type*} [Field 𝕜][LinearOrder 𝕜][IsStrictOrderedRing 𝕜][AddCommGroup E][AddCommGroup β][PartialOrder β][IsOrderedAddMonoid β][Module 𝕜 E] [Module 𝕜β][IsStrictOrderedModule 𝕜β] {s : Set E} {f : E →β} {t : Finset ι} {w : ι→ 𝕜} {p : ι → E} (hf :ConvexOn 𝕜 s f) (h₀ : ∀i ∈ t, 0 ≤ w i) (h₁ : ∑i ∈ t, w i = 1) (hmem :∀ i ∈ t, p i ∈ s) : f (∑i ∈ t, w i • p i) ≤ ∑ i∈ t, w i • f (p i)Statement map for Jensen's InequalityConvexity bounds the value at a normalized finite weighted sum by the corresponding weighted sum of the values. The pinned upstream declaration is ConvexOn.map_sum_le. The exact checked statement is theorem ConvexOn.map_sum_le {𝕜 E β ι : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [AddCommGroup β] [PartialOrder β] [IsOrderedAddMonoid β] [Module 𝕜 E] [Module 𝕜 β] [IsStrictOrderedModule 𝕜 β] {s : Set E} {f : E → β} {t : Finset ι} {w : ι → 𝕜} {p : ι → E} (hf : ConvexOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) : f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i).Mathematical readingConvexity bounds thevalue at a normalizedfinite weighted sum bythe correspondingweighted sum of thevalues.Pinned declarationmathlib ·ConvexOn.map_sum_leExact checked formtheoremConvexOn.map_sum_le {𝕜 Eβ ι : Type*} [Field 𝕜][LinearOrder 𝕜][IsStrictOrderedRing 𝕜][AddCommGroup E][AddCommGroup β][PartialOrder β][IsOrderedAddMonoid β][Module 𝕜 E] [Module 𝕜β][IsStrictOrderedModule 𝕜β] {s : Set E} {f : E →β} {t : Finset ι} {w : ι→ 𝕜} {p : ι → E} (hf :ConvexOn 𝕜 s f) (h₀ : ∀i ∈ t, 0 ≤ w i) (h₁ : ∑i ∈ t, w i = 1) (hmem :∀ i ∈ t, p i ∈ s) : f (∑i ∈ t, w i • p i) ≤ ∑ i∈ t, w i • f (p i)

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Nonnegative normalized contributions form one barycenter, and the convex value there lies no higher than the weighted combination of the four displayed values. Explanatory scientific diagram.
Detailed visual description

Inside a softly bounded lower domain, a circle, square, triangle, and diamond send forward blue strands of different positive widths into one braided green central knot. Matching outlined shapes sit on a charcoal upward-convex curve above. Four patterned blue paths converge from those curve points to a gold weighted-value marker, while a dashed vertical guide descends to a green marker on the curve directly above the barycenter. A pale gold echo ring around the lower marker keeps coincidence visually available, so the representative spacing does not turn the non-strict comparison into a strict theorem.

Why it matters

A mathematical landmark

Jensen's inequality is one of the central organizing principles of convex analysis, converting geometric convexity into a quantitative bound for averages. Mathlib's selected declaration gives a broad finite weighted-sum form over ordered modules, making the normalization and domain-membership hypotheses explicit.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's finite weighted-sum form of Jensen's inequality. The selected declaration assumes a convex function on s, finitely many points p i in s, nonnegative weights w i, and the exact normalization ∑ i ∈ t, w i = 1; it concludes f (∑ i ∈ t, w i • p i) ≤ ∑ i ∈ t, w i • f (p i). It is not integral or expectation Jensen, the strict inequality or equality case, the concave reverse inequality, or a special theorem for a fixed number of points.

Source and local evidence

Where the theorem comes from

Existing declaration
ConvexOn.map_sum_le in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.jensen-inequality.v001
Source
Open the pinned upstream reference