Strict and strong convexity

Definition

A function \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\) is

  • Strictly convex if, \(\forall x,y,x≠y,\forall λ∈(0,1):f(λx+(1-λ)y<λf(x)+(1-λ)f(y)\).

  • Strongly convex if, \(\exists α>0\) such that \(f(x)-α‖x‖²\) is convex.

Example

1) The function\( f(x)=x^{2}\) is strongly convex on \( \mathbb{R}\).

2) The function \(f(x)=x^{4}\) is strictly convex on \( \mathbb{R}\).

Proposition

Strong convexity implies Strict convexity implies Convexity.

Remark : But the converse of neither implication is true.

Proof

The fact that strict convexity implies convexity is obvious.

To see that strong convexity implies strict convexity, note that strong convexity of \(f\) implies

\(\hspace{1cm}f(λx+(1-λ)y)-α‖λx+(1-λ)y‖^{2}≤λf(x)+(1-λ)f(y)-αλ‖x‖^{2}-α(1-λ)‖y‖^{2}\).

But

\(\hspace{2cm}λα‖x‖^{2}+(1-λ)α‖y‖^{2}-α‖λx+(1-λ)y‖^{2}>0,\forall x,y,x≠y,\forall λ∈(0,1)\)

because \(‖x‖^{2}\) is strictly convex. The claim follows.

To see that the converse statement are not true, observe that \(f(x)=x\) is convex but not strictly convex and \(f(x)=x^{4}\) is strictly convex but not strongly convex.