Take C to be a closed convex set in $l^p$ (the space of sequences equipped with the $p$-norm where p>1) such that:


i) 0 is in C


ii) C is strictly larger than 0


iii) $C \cap -C =\{0\}$


iv) $C \subset B(0,1)$, that is, all members of $C$ have norm less than or equal to 1


Then, is there a non-trivial linear functional $q$ such that $q \cdot c \geq 0$ for all $c \in C$ and $q \cdot d > 0$ for some $d \in C$?


Edit: If the cone generated by C is closed, then Klee (1955) Theorem 2.7 proves that there is a q such that $q \cdot c>0$ for all non-zero c, an even stronger result than the one I am looking for.


<https://pdfs.semanticscholar.org/1322/2f3946fcc565c6263f6e31c2ae0cb2e92ad7.pdf>



Supporting Hyperplane Theorem in Lp Spaces
Let $C'= \bigcup\_n nC$ be the cone generated by $C$ and let $X$ be the closed linear span of $C$.


Theorem: A bounded linear functional $q$ with the stated property exists if and only if $C'$ is not dense in $X$.


Proof: Any linear functional which is nonnegative on $C$ will also be nonnegative on $C'$, so if $C'$ is dense in $X$ the only bounded linear functional which is nonnegative on $C$ is the zero functional. Conversely, suppose $C'$ is not dense in $X$ and let $x \in X \setminus \overline{C'}$. Then $\overline{C'}$ is a closed convex set containing the origin and by a standard separation theorem there is a bounded linear functional $q$ and a constant $a < 0$ such that $q(x) < a$ and $q(y) \geq a$ for all $y \in \overline{C'}$. But if $q(y) < 0$ for any $y \in \overline{C'}$ then $q(ny) < a$ for sufficiently large $n$, a contradiction; thus $q$ is nonnegative on $\overline{C'}$. Since $C$ generates $X$ and $q$ is not zero on $x \in X$, it cannot vanish on $C$. ///


Can $C'$ ever be dense in $X$? I think so, even for $p = 2$. Let $(e\_n)$ be the standard basis of $l^2$. For each $n$ let $\alpha\_n = 2^{-n}$. Then, for each $n$ let $(x^n\_k)$ be a sequence in the unit sphere of ${\rm span}(e\_1, \ldots, e\_n)$ which hits each element of a dense subset of that sphere infinitely many times. For each $n$ and $k$ let $m = 2^n(2k + 1)$ and define $v^n\_k = \alpha\_m x^n\_k + \alpha\_m^2 e\_m$. Thus, $v^n\_k$ is a slight perturbation of a small multiple of $x^n\_k$, and all the perturbations are in different coordinate directions. It seems to me that the closed convex hull $C$ of all the $v^n\_k$ satisfies $C \cap -C = \{0\}$, yet the cone $C$ generates has every $x^n\_k$ in its closure (it contains $x^n\_k + \alpha\_m e\_m$, and the same vector $x^n\_k$ appears infinitely many times in the $n$th sequence, therefore with arbitrarily large $m$) and hence it is dense in $l^2$. Proving the first part might take a little work, though.


