# Convex Polytope

My Linear Programmation book tells me to consider the closed and limited Convex Polytope defined as

$$S=\left \{ \right.\bar{x}:A\bar{x}=\bar{b} ,\bar{x}\geq 0\left. \right \}$$

But , does this equation only define the points that belongs to the hyperplanes that contains the convex polytope ? Thanks

• It is a strict equality $A\overline x = \overline b$ and no inequality so any solutions must lie on planes or lines and can't lie in the "volume" or what you would call it. – mathreadler Jul 22 '17 at 12:33
• So , am i right ? – JDOE Jul 22 '17 at 12:47
• I don't know. I am not sure I understand your question. – mathreadler Jul 22 '17 at 13:41
• A polytope is the set of all points included in a system of hyperplanes or just the points in the hyperplanes ? Because my book calls polytope the set i wrote in the main post – JDOE Jul 22 '17 at 13:44
• $S$ is the set of all $\overline x$ which fulfil both the equality and the inequality. I don't get the question. It seems you are more confused about semantics. I don't know how your book defines those words. – mathreadler Jul 22 '17 at 14:06

## 1 Answer

This field is not my meat, but let me try to answer according to my understanding.

This is the intersection of the $(+,+,\dots,+)$-cone of your domain space with a (translated) linear subspace thereof. For instance, if the domain space is just the plane, you’re talking about either a single point in the first quadrant, or the first-quadrant part of a line, or all the first quadrant.

If the domain space is $\Bbb R^3$, you’re looking either at a single point in the first octant, or the first-octant part of a line, or the first-octant part of a plane, or the all the first octant.

I don’t know what the book means by “limited”: if they mean compact, that is certainly not the case; if they mean closed, that is all right.

The set defined here turns out to be convex, but you are making a mistake if you are trying to see this definition as a modification of the book’s definition of a convex set. I would say that there are no hyperplanes specifically mentioned here, though if you search for them, you will find them.