\begin{definition}[Definition:Quasi-Concave Total Preordering]
Let $R^n$ be a real vector space.
Let $\precsim$ be a total preordering on $\R^n$.
Let $\precsim$ be such that:
:for all $b \in \R^n$, the set $\left\{ {a \in \R^n: b \precsim a}\right\}$ is convex.
Then $\precsim$ is '''quasi-concave'''.
\end{definition}