Concavity (Convexity)

Definition. A set $S$ is said to be concave if for every vector $x$ and $y$ in $S$, $\lambda x + (1-\lambda) y$ is in $S$ for all $0\leq \lambda \leq 1$. In other words, all points on the line between two points of $S$ lie in $S$.

Definition. A functional is a mapping from a vector space to the real numbers $\Re$.

Thus, for example, every norm is a functional.

Definition. A linear functional is a functional $f$ such that for each $x$ and $y$ in the linear space $X$, and for all scalars $\alpha$ and $\beta$, we have $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$.

Definition. The norm of a linear functional $f$ is defined on the normed linear space $X$ by

$$\left\Vert\,f\,\right\Vert \mathrel{\stackrel{\mathrm{\Delta}}{=}}\sup_{x\in X} \frac{\left\vert f(x)\right\vert}{ \left\Vert\,x\,\right\Vert}.$$

Definition. A functional $f$ defined on a concave subset $S$ of a vector space $X$ is said to be concave on $S$ if for every vector $x$ and $y$ in $S$,

$$\lambda f(x) + (1-\lambda) f(y) \geq f\left(\lambda x + (1-\lambda) y\right) ,\qquad 0\leq \lambda \leq 1.$$

A concave functional has the property that its values along a line segment lie below or on the line between its values at the end points. The functional is strictly concave on $S$ if strict inequality holds above for $\lambda\in(0,1)$. Finally, $f$ is uniformly concave on $S$ if there exists $c>0$ such that for all $x,y\in S$,

$$\lambda f(x) + (1-\lambda) f(y) - f\left(\lambda x + (1-\lambda) ... ...\lambda(1-\lambda)\left\Vert\,x-y\,\right\Vert^2 ,\qquad 0\leq \lambda \leq 1.$$

We have

$$\hbox{Uniformly}\;\; \hbox{Concave}\quad\implies\quad \hbox{Strictly}\;\; \hbox{Concave}\quad\implies\quad \hbox{Concave}$$

Definition. A local minimizer of a real-valued function $f(x)$ is any $x^\ast$ such that $f(x^\ast)<f(x)$ in some neighborhood of $x$.

Definition. A global minimizer of a real-valued function $f(x)$ on a set $S$ is any $x^\ast\in S$ such that $f(x^\ast)<f(x)$ for all $x\in S$.

Definition. A cluster point $x$ of a sequence $x_n$ is any point such that every neighborhood of $x$ contains at least one $x_n$.

Definition. The concave hull of a set $S$ in a metric space is the smallest concave set containing $S$.