Chapter 15

Show that any hyperplane has the form \(\mathcal{H}\left(N,\|N\|^{2}\right)\) for an appropriate vector \(N\).

If \(A\) is an \(m \times n\) matrix prove that the set \(\left\\{A x \mid x \in \mathbb{R}^{n}, x>0\right\\}\) is a convex cone in \(\mathbb{R}^{m}\).

If \(A\) and \(B\) are strictly separated subsets of \(\mathbb{R}^{n}\) and if \(A\) is finite, prove that \(A\) and \(B\) are strongly separated as well.

Let \(V\) be a vector space over a field \(F\) with char \((F) \neq 2\). Show that a subset \(X\) of \(V\) is closed under the taking of convex combinations of any two of its points if and only if \(X\) is closcd under the taking of arbitrary convex combinations, that is, for all \(n \geq 1\), $$ x_{1}, \ldots, x_{\mathrm{n}} \in X, \sum_{i=1}^{m} r_{i}=1,0 \leq r_{i} \leq 1 \Rightarrow \sum_{i=1}^{n} r_{i} x_{i} \in X $$

Explain why an \((n-1)\)-dimensional subspace of \(\mathbb{R}^{n}\) is the solution set of a linear equation of the form \(a_{1} x_{1}+\cdots+a_{n} x_{n}=0\).

Prove that the convex hull of a \(\operatorname{set}\left\\{x_{1}, \ldots, x_{n}\right\\}\) in \(\mathbb{R}^{n}\) is bounded, without using the fact that it is compact.

Suppose that a vector \(x \in \mathbb{R}^{n}\) has two distinct representations as convex combinations of the vectors \(v_{1}, \ldots, v_{n}\). Prove that the vectors \(v_{2}-v_{1}, \ldots, v_{n}-v_{1}\) are linearly dependent.

Suppose that \(C\) is a nonempty convex subset of \(\mathbb{R}^{n}\) and that \(\mathcal{H}(\mathrm{N}, b)\) is a hyperplane disjoint from \(C\). Prove that \(C\) lies in one of the open half-spaces determined by \(\mathcal{H}(N, b)\).

Find two nonempty convex subsets of \(\mathbb{R}^{2}\) that are strictly separated but not strongly separated.

Prove that \(X\) and \(Y\) are strongly separated by \(\mathcal{H}(N, b)\) if and only if $$ \left.\left\langle N, x^{\prime}\right\rangle\right\rangle b \text { for all } x^{\prime} \in X_{e} \text { and }\left\langle N, y^{\prime}\right\rangle