Theorem 13.6.5.label Let $A\suf\R^{d}$ be a convex set with $int(A)=\emp$ then there exists an affine subspace $L\suf\R^{d}$ such that $A\suf L$ and $\dim L<d$.
Proof. Let $k$ be the maximum number of affinely independent points in $A$. By Lemma 13.6.4, $k<d+1$. Let $\curl{v_i }_{1}^{k}$ be affinely independent points then for each $v\in A$ there exists a solution $\gam\neq 0$ to
\begin{align*}\sums{k }{i=1}\gam_{i} v_{i}+\gam v=0&&\sums{k }{i=1}\gam_{i}+\gam=0\end{align*}
Then $v\in A$ can be expressed as an affine combination
\begin{align*}v=\sums{k }{i=1}\parens{-\frac{\gam_{i}}{\gam}}v_{i}\end{align*}
Showing $A$ is contained in the affine subspace $L\define aff(\curl{v_i}_{1}^{k})$ so $\dim L=k-1<d$.$\square$
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