If the interior of two convex manifolds intersect, what is the dimension of their intersection?

Question

In $\mathbb R^n$, let $A,B$ be two convex manifolds of dimension $a,b$ respectively whose interiors intersect. It would seem that their intersection is a convex manifold of dimension $\min(a,b)$. Is that correct?

What if we drop some of the conditions? For example, this would seem to hold true even if $A,B$ are not manifolds, but simply complex sets. Would it hold true outside of $\mathbb R^n$?

Answer 1

Drawing on Moishe Kohan's suggestion, we first prove the following:

Prove that if $\mathcal V$ is a finite dimensional real vector space, $\mathcal U, \mathcal W \subset \mathcal V$ are linear subspaces whose common span is $\mathcal V$ then $\operatorname{codim}(\mathcal U \cap \mathcal W)=\operatorname{codim} \mathcal U+ \operatorname{codim}\mathcal W$, where $\operatorname{codim}$ denotes codimension in $\mathcal V$.

For simplicity of discussion, we take $\mathcal V$ for the ambient space. If $\mathcal A$ is a vector space, there exists a matrix $A$ whose rowspace $R(A)$ is precisely $\mathcal A$. Then $\operatorname{codim} \mathcal A = \dim N(A)$.

Recall that if $A, B$ are two vector spaces such that $A \cap B = \{0\}$, and $C = \operatorname{span}(A, B)$, then $\dim C = \dim A + \dim B$.

Since $\mathcal U, \mathcal W$ span $\mathcal V$, then $N(U) \cap N(W) = \{0\}$, so $\dim N(U) \cap N(W) = \dim N(U) + \dim N(W)$. Generalizing this result to cases where $\mathcal V$ is not the ambient space but rather an arbitrary finite dimensional real vector space is straightforward.

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This suggests the following: Let $A, B$ be two convex sets in $\mathbb R^n$ whose interiors intersect. Choose an arbitrary point in this interior intersection and call it $O$. Let $V$ be the set of vectors expressible as $O + \lambda_1 (a - O) + \lambda_2 (b - O)$ for any $a \in A, b \in B$. Then $$\operatorname{codim} A \cap B = \operatorname{codim} A + \operatorname{codim} B$$ where $\operatorname{codim}$ is in $V$.