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Consider the normed vector space $E=\mathbb{R}^n$. Define
$ P=\{x \in \mathbb{R}^n: x_i \geq 0, \forall i \}$.
Let $M$ be a subspace such that $M \cap P = \{0\}$. I want to see that $M^\perp \cap {\rm Int}(P) \neq \emptyset $. This seems obvious geometrically, any idea how a short proof would look like?
Here is a proof based on another fact that seems geometrically obvious, and which I will prove below.
Lemma For any set of vectors $v_1,\ldots,v_n$ in a real vector space $V$ for which there exists no non-trivial relation $\vec0=\alpha_1 v_1+\cdots+\alpha_n v_n$ with $\alpha_i\geq0$ for all $i$, there exists a linear form $f$ on $V$ such that $f(v_i)>0$ for all $i$. Under the weaker hypothesis that there exists no relation $\vec0=\alpha_1 v_1+\cdots+\alpha_n v_n$ with $\alpha_i>0$ for all $i$, there exists a linear form $f$ on $V$ such that $f(v_i)\geq0$ for all $i$ and $f(v_i)>0$ for at least one $i$.
Conversely, in both cases, such a linear form immediately witnesses the impossibility of relations of the indicated kind.
Now suppose $M$ is a subspace with $M\cap P=\{\vec0\}$, and, for a contradiction, also $M^\perp\cap\mathrm{Int}(P)=\emptyset$. From the first fact we see that the $n$ images of the canonical basis $\{e_1,\ldots,e_n\}$ of $\mathbb R^n$ orthogonally projected onto $M^\perp$ satisfy the conditions of the first part of the lemma: if there were a non-trivial relation with non-negative coefficients between those projections, then the corresponding non-trivial linear combination of the vectors $e_i$ themselves would lie in $M\cap P$ without being $\vec0$. So the lemma says there is a linear form on $M^\perp$ with strictly positive values on those projections. It can be realised as the scalar product with a vector $v\in M^\perp$, which therefore has $\langle v \mid e_i \rangle>0$ for all $i$; in other words, $v=(v_1,\ldots,v_n)$ with $v_i>0$ for all $i$.
Similarly, $M^\perp\cap\mathrm{Int}(P)=\emptyset$ means that the orthogonal projections of the basis vectors $e_i$ onto $M$ admit no relation between them with all coefficients strictly positive, since the corresponding linear combination would lie in $M^\perp\cap\mathrm{Int}(P)$. By the second part of the lemma this means that there is a non-zero linear form on $M$ that takes non-negative values on all those projections. It can be realised as the scalar product with a vector $w\in M$, which therefore has $\langle w \mid e_i \rangle\geq0$ for all $i$; in other words, $w=(w_1,\ldots,w_n)$ with $w_i\geq0$ for all $i$ and $w_i>0$ for at least one $i$. But then $\langle v\mid w\rangle=v_1w_1+\cdots+v_nw_n>0$, contradicting $v\in M^\perp$ and $w\in M$ so that $v\perp w$.
Now for the proof of the lemma. In both statements we may assume without loss of generality that $V$ is the span, as a vector space, of the vectors $v_1,\ldots,v_n$: that case will provide us with a linear form on the span of those vectors, which can then be arbitrarily extended to the whole space. We shall use the fundamental fact that the cone of the non-negative linear combinations of some finite set of vectors can also be described as the cone of vectors where some finite set of linear forms all take non-negative values (the Minkowski-Weyl theorem).
The hypothesis in the first part of the lemma implies that the cone $C$ of non-negative linear combinations of $v_1,\ldots,v_n$ does not contain any subspace of dimension${}>0$: if it did one could add the combinations giving two opposite non-zero vectors in the subspace to get a non-trivial relation with non-negative coefficients. This means that no vector vanishes under all the linear forms that define $C$, and in particular none of the vectors $v_i$ do. But then taking the sum of all these linear forms gives a linear form that takes a strictly positive value on all the vectors $v_i$, proving this part.
Under the weaker hypothesis of the second part of the lemma, $C$ may contain a subspace of positive dimension, but not the whole space: in that case every $v_i$ would occur in some relation with non-negative coefficients (found using a non-negative expression for $-v_i$), and adding them up would give a relation with all strictly positive coefficients. Then the set of linear forms defining $C$ contains at least one linear form that takes a non-zero value on at least one $v_i$. It takes non-negative values on all of $C$ and in particular on every vector $v_i$, as required.
This is an elementary proof I discovered using a geometric version of Hahn-Banach. Hope you like it. Thanks to all of you who contributed to the discussion. :)
Suppose $M^\perp \cap \rm{Int}P=\emptyset$. Then since both sets are convex and $\rm{Int}P$ is open, it follows that there exists $0 \neq f \in E^*$ that separates them. That is, there exists $\alpha \in \mathbb{R}$ such that
$f(x) \leq \alpha \leq f(y) \text{ for all } x\in M^\perp, y \in \rm{Int} P. $
By Reisz's Lemma $\exists x^* \in E$, $x^*\neq0$, such that $f(x)=(x,x^*)$. Now it is easy to see that since $M^\perp$ is a subspace, the first inequality implies that $(x,x^*)=0$ for all $x \in M^\perp$, ie $x^* \in (M^\perp)^\perp=M$ (since $E$ is finite dimensional).
Therefore, from the second inequality we have $(y,x^*) \geq 0$ for all $y\in \rm{Int}P$. Finally, one only needs to note that $e_i \in \overline{\rm{Int}P}$ for all $i=1,\cdots,n$. The continuity of the inner product now implies that $(e_i,x^*) \geq 0$ for all $i$. I.e., $x^* \in P \cap M$ and $x^* \neq 0$, a contradiction. $\square$
Based on the comments of my previous answer, I realise that one can simplify the argument considerably. First of all, if $M$ is a hyperplane, then the result is easily proved as follows. One can define $M$ by a linear equation $M=\{(x_1,\ldots,x_n)\in\mathbb R^n\mid x_1a_1+\cdots+a_nx_n=0 \}$ where one can take at least one of the coefficients $a_i$ to be positive. This being so, no other $a_j$ can be non-positive, since otherwise taking $x_i=-a_j$, $x_j=a_i$ and all other coordinates $x_k=0$ would give a non-zero vector in $M\cap P$. But if all $a_j$ are positive, one has $(a_1,\ldots,a_n)\in M^\perp\cap \mathrm{Int}(P)$.
Now what remains is to show that any subspace $M$ intersecting $P$ in $\vec0$ only can be extended to a hyperplane $H$ with the same property, because it then follows that $M^\perp\supseteq H^\perp$ which as we have seen meets $\mathrm{Int}(P)$. Projecting onto $M^\perp$, we need to find an appropriate image for $H$; this needs to be a hyperplane $H'$ of $M^\perp$ that meets the image $\cal C$ of $P$ (by orthogonal projection) only in $\vec0$. But since $P$ meets $M$ only in $\vec0$, its image $\cal C$ is a convex cone that is disjoint from $\mathrm{Int}(-\cal C)$. But the Hahn-Banach separation theorem (on OP's request) provides a linear form that separates $\cal C$ and $\mathrm{Int}(-\cal C)$, and its kernel provides the requested $H'$ (this could also be done using the first part of the lemma in my other answer). Finally take $H=H'\oplus M$.
