Convex and strictly convex functionals on a convex subset of a real vector space

Let $X$ be a real vector space and $K\subseteq X$. The set $K$ is convex if $\lambda u+(1-\lambda )v\in K$ for all $u,v\in K$ and $\lambda \in [0,1]$. Fix a nonempty convex $K$ and an extended-real functional $I:K\to (-\mathrm{\infty },+\mathrm{\infty }]$ (Proper, coercive and weakly lower semicontinuous extended-real functionals), with the sums and positive-weight products of The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined. For convex combinations only, additionally define $0\cdot (+\mathrm{\infty })=0$; this is a local convention, since that product is left undefined in the general extended-real arithmetic. Then $I$ is convex if $I(\lambda u+(1-\lambda )v)\le \lambda I(u)+(1-\lambda )I(v),u,v\in K,\lambda \in [0,1],$ and strictly convex if the inequality is strict whenever $u\mathrm{\ne }v$, $I(u),I(v)<+\mathrm{\infty }$ and $\lambda \in (0,1)$. A convex functional has convex sublevel sets: for every $t\in \mathbb{R}$ the set $\{u\in K:I(u)\le t\}$ is convex. Only real coefficients are used: on a complex vector space these notions are read on the underlying real structure.

• Sublevel sets. Let $I$ be convex and let $t\in \mathbb{R}$. If $u,v\in K$ satisfy $I(u)\le t$ and $I(v)\le t$, then $I(u),I(v)<+\mathrm{\infty }$, and for $\lambda \in [0,1]$ convexity and the extended-real conventions give $I(\lambda u+(1-\lambda )v)\le \lambda I(u)+(1-\lambda )I(v)\le \max \{I(u),I(v)\}\le t$; hence $\{u\in K:I(u)\le t\}$ is convex. This is the property used when a sublevel set is intersected with a weakly closed admissible set.

• Endpoint coefficients. At $\lambda =0$ and $\lambda =1$ the defining inequality reads $I(v)\le I(v)$ and $I(u)\le I(u)$, using $0\cdot (+\mathrm{\infty })=0$ for the extended value $+\mathrm{\infty }$; the strict form is therefore imposed only for $0<\lambda <1$, as stated.

• Finite competitors. For $0<\lambda <1$, if either $I(u)$ or $I(v)$ is $+\mathrm{\infty }$, the right-hand side of the convexity inequality is $+\mathrm{\infty }$, so it carries no information at such a pair; the strict form is correspondingly restricted to pairs in the effective domain $\mathrm{dom}I$ (Proper, coercive and weakly lower semicontinuous extended-real functionals).