DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

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A supporting line of slope $m$ for a real function at an interior point

Definition

Let $f:I\to\mathbb R$ , where $I$ is an interval (Intervals of $\mathbb{R}$ : the nine order-convex forms, nondegeneracy, and length), and let $c$ be an interior point of $I$ . A line of slope $m$ supports $f$ at $c$ when

$f(x)\ge f(c)+m(x-c)$

for every $x\in I$ . The supporting line is the affine function $x\mapsto f(c)+m(x-c)$ .

Depends on

Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length

Used by

Every slope between the left and right derivatives of a convex function gives a supporting line Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 8 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. Gardner, Convex Functions, Notes 6.6 (standard reference, not scraped)

A supporting line of slope mmm for a real function at an interior point