partial ordering induced by a cone

Question

In this question, it is said that a cone $K$ induces (1) a partial ordering on a set $S$; and (2) a set of generalized inequalities.

What does it mean for a cone to induce a partial ordering on a set? Is there any geometric intuition for this?

Answer 1

Update: p.84 in Dattorro's CVX book provides a nice explanation along with a great visual on p.146.

From what I gather, a cone $K$ induces a partial order by specifying which particular points can be compared (ordered). Points comparable to a particular point $x$ are those points $C$ that fall within the cone $K$ shifted to vertex $x$.

Answer 2

Cannot comment, but note that you can also define a generalized inequality w.r.t. to a point not in the original cone. For example, all points less than some particular point $y$ belong to the cone $y - K$ if $K$ is the original cone (and we can say $y \succeq x$ for $x \in w-K$).