Codimension of the largest subspace of a closed convex cone in a normed vector space Please, help to prove (or disprove) the following statement.
Let $K \neq V$ be a closed convex cone ($K+K=K$, $\alpha K \subseteq K$ for any $\alpha \geq 0$) in a real normed vector space $V$. If $-K \cup K = V$, then the subspace $-K \cap K$ has codimension 1.
Thanks.

An attempt of proof. Please, check.
Let $X:=-K \cap K$. Fix $y_0 \in K\setminus X$. The task is to prove $X+ \mathbb R y_0=V$.
Claim: given $v \in V$, there exist either $\inf \left\{ \beta \in \mathbb R: v+ \beta y_0 \in K \right\}$ or $\sup \left\{ \beta \in \mathbb R: v+ \beta y_0 \in K \right\}$. Indeed, if both do not exist, then $v+ \beta y_0 \in K$ for any $\beta \in \mathbb R$. In particular, $n^{-1}v+y_0 \in K$ and $n^{-1}v-y_0 \in K$ for any natural $n$. Tending $n$ to infinity, this implies a contradiction: $y_0 \in X$ (since $K$ is closed). Let $\alpha$ be the existed (lower or upper) bound. Then $v+ \alpha y_0 \in \partial K$. But $\partial K=X$, since $K$ is closed and $-K \cup K = V$. This completes the proof.
