# Explain `All polyhedrons are convex sets´

My teacher in course in Mat-2.3140 of Aalto University claims that 'All polyhedrons are convex sets' here. This premise was in a false-or-not-problem 'The feasible set of linear integer problem is polyhedron'. You can see below the screenshot of the solution.

Wikipedia shows nonconvex polyhedrons such as orthogonal polyhedron here.

What should I now believe? Is polyhedron convex or not?

Definitions on the lecture slides (p.8, L4)

Polyhedron is such that $$P=\{\bar x\in \mathbb R^n | A \bar x\geq \bar b\}, A\in \mathbb R^{m\times n},\bar b\in \mathbb R^m$$ and a convex function $f(x)$ must satisfy $$f(\lambda \bar x+(1-\lambda)\bar y)\leq \lambda f(\bar x)+(1-\lambda) f(\bar y) \text{, } \forall \bar x, \bar y, \lambda \in [0,1]$$ and a convex set $C$ is such that $$\bar x, \bar y \in C\rightarrow \lambda \bar x+(1-\lambda)\bar y\in C.$$

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Just so you know, your first link is only available to Aalto students. As to your question - does the course contain a definition of polyhedron? It may simply be a question of two different definitions, one allowing non-convex things and the other not. – Matthew Pressland Jan 11 '13 at 10:46
@MattPressland shared relevant information in the q. – hhh Mar 4 '13 at 14:34
In convex optimization, the word "polyhedron" is sometimes used in a way that is inconsistent with the way the word is defined in basic geometry. – littleO Jun 21 at 9:52

I suspect you are confused with the definition. Usually a a polyhedron is defined by specifying a finite subset of $n-1$ dimensional affine subspaces in $\mathbb{R}^{n}$. In this way what you get is always convex. This is the definition people use when work on combinatorical topology or algebraic combinatorics. You should confirm this with your teacher though.

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Is this "[u]sually a polyhedron is defined by specifying a finite subset of $n-1$ dimensional affine subspaces in $\mathbb R^n$" the method Ross uses here? – hhh Feb 26 '13 at 22:45
Is there anything like concave polyhedrons? What is the name for concave things such as ball-with-hole, cube-with holes and tetrahedron with a dent if not polyhedrons? Torus, n-torus but the other things? Some general name? – hhh Mar 4 '13 at 14:35
@hhh: I do not know what you are really talking about. – Bombyx mori Mar 4 '13 at 18:49
@hhh: They are called non-convex polyhedra, if they have faces that are polygons (bounded by straight line segments.) The faces of non-convex polyhedra can either be convex polygons, star-polygons (like the pentagram), or skew polygons (which don't lie in a plane.) "Star-polyhedra" are a particular type of non-convex polyhedra. Things like balls and tori are not polyhedra. Those are called surfaces of nonzero genus. (The genus is the number of holes.) – Kundor Apr 9 '14 at 2:27

Look at Boyd's book Section 2.2.4 http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

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A polyhedron can be defined as a finite intersection of halfspaces and hyperplanes. Halfspaces and hyperplanes are convex sets, the intersection of convex sets is a convex set, and thus all polyhedrons are convex.

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