When is the intersection of three cones connected?

Question

I am interested in the set of points $x \in \mathbb{R}^n$ described by $Ax \ge 0$, $Bx \ge c$, and $x > 0$. $A$ and $B$ are $n \times n$ matrices, and $c \in \mathbb{R}^n$. I would like to know under what circumstances this set is connected.

Thanks for any advice.

What does $a\geq b$ mean when $a, b \in \Bbb R^n$? – Arthur Oct 05 '12 at 09:23 — Arthur, Oct 05 '12 at 09:23

martini · Accepted Answer · 2012-10-05T09:42:55.937

0

As $\{x \mid Ax \ge 0\}$, $\{x \mid Bx \ge c\}$ and $\{x \mid x > 0\}$ are convex, their intersection is also convex and hence connected.

edited Oct 05 '12 at 09:42

answered Oct 05 '12 at 09:30

martini

84,101

When is the intersection of three cones connected?

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