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Given $P = \{x \in\mathbb R^n \mid a_1x_1 + \ldots + a_nx_n = \text{constant}\}$, $(a_1, \ldots , a_n) \ne 0$. Can $P$ be a polytope? I think that with $N = 1$, $P$ is a point. Can a point in $\mathbb R^1$ be a polytope?

Thank you all!

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  • $\begingroup$ $P$ looks more like a hyperplane. $\endgroup$
    – Earthliŋ
    Jun 17, 2013 at 15:41
  • $\begingroup$ What if N = 1 ? $\endgroup$ Jun 17, 2013 at 15:57
  • $\begingroup$ Then you have a 0-dimensional hyperplane, i.e. a point, which you can interpret as a 0-dimensioal polytope if you like. Your formula for $P$ only gives you polytopes when $n=1$, though. For $n>1$, the formula defines a hyperplane, which won't be a polytope. $\endgroup$
    – Earthliŋ
    Jun 17, 2013 at 16:10
  • $\begingroup$ Many issues related to polytopes are discussed in Branko Grunbaum's excellent book Convex Polytopes. $\endgroup$ Jun 17, 2013 at 16:33

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This is essentially an issue of semantics. There are two ways of defining polytopes: the V-representation (the convex hull of a finite set of points) and the H-representation (the intersection of half-spaces).

V-representations are always compact, but H-representations don't have to be. A lot of people would reserve the word "polytope" for these objects when they are compact. Can you see how $P$ is the intersection of two halfspaces?

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