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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.

enter image description here

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. –  Matt Pressland Jan 11 '13 at 10:46
    
@MattPressland shared relevant information in the q. –  hhh Mar 4 '13 at 14:34
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2 Answers 2

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
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@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 at 2:27
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Look at Boyd's book Section 2.2.4 http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

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