Convex and Concave Functions on $\mathbb{R}^n$¶

Definition¶

Let $C\subseteq\mathbb{R}^n$ be convex. A function $f\colon C\to\mathbb{R}$ is convex if for all $\boldsymbol{x},\boldsymbol{y}\in C$ and all $\lambda\in[0,1]$$[0,1]$,

It is concave if the inequality is reversed.

Interpretation¶

A convex function lies below the straight interpolation of its values. A concave function lies above that interpolation.

The two graphs illustrate the segment test: on a convex domain $C\subseteq\mathbb{R}^n$${\mathbb{R}}^n$, apply the defining inequality to every segment joining $\boldsymbol{x},\boldsymbol{y}\in C$$C$. Reversing the inequality gives concavity; equivalently, $f$ is concave if and only if $-f$ is convex. For a twice continuously differentiable function on an open convex domain, the Hessian test checks positive semidefiniteness for convexity and negative semidefiniteness for concavity.

A convex domain keeps the interpolated input inside the domain, so its function value is defined. The convexity inequality then compares this output with the same weighted interpolation of the endpoint outputs.

Key Results¶

• For convex functions, every local minimum is a global minimum.
• For concave functions, every local maximum is a global maximum.
• If $f$ is strictly convex, it has at most one minimizer on $C$; if $f$ is strictly concave, it has at most one maximizer on $C\text{.}$

Examples¶

• The function $f(x,y)=x^2+y^2$ is convex on $\mathbb{R}^2$.
• The function $u(x,y)=\log x+\log y$ is concave on $(0,\infty)^2$.

Prerequisites¶

• Convex Set in $\mathbb{R}^n$
• Function

Related Concepts¶

• Hessian Test for Convexity and Concavity
• Global Maximum and Minimum on $\mathbb{R}^n$
• Convexity and Concavity

Sources¶

• Math IV: Multivariate Calculus · Week 4, Lecture 1

Keywords: convex function on Rn, concave function on Rn, multivariable convexity Symbols: element of, infinity, lambda, real numbers, subset

Last updated: 27 October 2026 11:52