Definition:Convex Real Function/Definition 1/Also presented as

Convex Real Function: Definition 1: Also presented as

By setting $\alpha = \lambda$ and $\beta = 1 - \lambda$, this can also be written as:

$\forall x, y \in I, x \ne y: \forall \lambda \in \openint 0 1: \map f {\lambda x + \paren {1 - \lambda} y} \le t\lambda \map f x + \paren {1 - \lambda} \map f y$

Geometric Interpretation

Let $f$ be a convex real function.

Then:

for every pair of points $A$ and $B$ on the graph of $f$, the line segment $AB$ lies entirely above the graph.

Also known as

A convex function can also be referred to as:

a concave up function

a convex down function.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 22$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convex function

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• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 12$