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I am trying to follow Roger Webster's Convexity 's proof of Euler's celebrated result on the relationship between the number of faces of a polytope. An image of the proof is here.

In the course of the proof, he states (without proof) that the intersection of an r-dimensional polytope (polytope is the convex hull of a finite number of points) in its relative interior with a hyperplane (an n-1 dimensional flat in R^n) gives rise to an r-1 dimensional polytope. This is highlighted in red in the image.

I am having difficulty trying to prove this although intuitively this seems obvious.

There are two things needed to be done here:

(1)To show that the intersection is a polytope, and,

(2)To show that the intersection is of dimension r-1

Regarding (2), what is proven before in the text is that the intersection of a hyperplane with a flat A of dimension r where the hyperplane meets but does not contain the flat is of Dim A - 1, i.e., r - 1. However, it is unclear to me how this can be used in this case.

A flat is essentially a set such that all affine combinations of points belongs to the set. i.e., if x,y\in A, then A is flat means that lambda x + (1-lambda) y \in A for all values of lambda.

I would be grateful for any help in trying to prove (1) and (2)

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    $\begingroup$ Letting $x_1,...,x_n$ be coordinates on $\mathbb{R}^n$, I believe a convex polytope of dimension $n$ is simply a compact region with nonempty interior defined by inequalities $0 \le f$ where $f$ is an affine function of the $x_i$ (please correct me if I am wrong on this definition). If you take the hyperplane to be $x_1 = 0$, it seems clear that these conditions are still satisfied by the intersection-- the intersection is obviously compact, has nonempty interior in $\mathbb{R}^{n-1}$ (unless it degenerates to a single point or an empty set), and is defined by affine inequalities. $\endgroup$
    – Bma
    Commented Apr 25, 2022 at 12:57
  • $\begingroup$ @Bma It is indeed true that every polytope can be represented as a convex hull and equivalently as the intersection of a finite number of closed halfspaces. Perhaps this is a good way to prove the result. Let me think about it. Thanks for your inputs. $\endgroup$
    – Tryer
    Commented Apr 25, 2022 at 13:09

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