Kerner’s breakdown minimization principle

An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way (as seen in the first example below) the order unit generalizes the unit element in the reals.

Definition

For the ordering cone $K\subseteq X$ in the vector space $X$, the element $e\in K$ is an order unit (more precisely an $K$-order unit) if for every $x\in X$ there exists a ${\lambda }_x>0$ such that ${\lambda }_{x}e-x\in K$ (i.e. $x{\le }_K{\lambda }_{x}e$).

Equivalent definition

The order units of an ordering cone $K\subseteq X$ are those elements in the algebraic interior of $K$, i.e. given by $\mathrm{core}(K)$.

Examples

Let $X=\mathbb{ℝ}$ be the real numbers and $K={\mathbb{ℝ}}_{+}=\{x\in \mathbb{ℝ}:x\ge 0\}$, then the unit element $1$ is an order unit.

Let $X={\mathbb{ℝ}}^n$ and $K={\mathbb{ℝ}}_{+}^n=\{x\in \mathbb{ℝ}:\mathrm{\forall }i=1,\dots ,n:x_i\ge 0\}$, then the unit element $\overrightarrow{1}=(1,\dots ,1)$ is an order unit.

References

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