DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-02

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

Definition

Let $f : I \to R$ , where $I$ is an interval (Intervals of $R$ : 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

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

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 · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

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