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Let $(V, \lVert \cdot \rVert)$ be a normed space. Let us consider the set $C = [-1,1]^{\dim V}$. The boundary of this set consists of closed subsets $B_i$ (indexed by some set $I$) of affine hyperplanes (the family of whose we will call $B$). Now, consider the set $K$ with the following properties:

  • $K$ is compact and convex,
  • $\forall i \in I\quad B_i \cap K \ne \emptyset$.

Question 1:

I would like to know whether from these properties of $K$ it follows that $0 \in K$.


It seems to be true if $\dim V < \infty$, with cases $\dim V \in \{ 1,2\}$ being rather easy to visualize. However, I wasn't able to find a simple argument which wouldn't result in checking many cases...


Question 2:

I'm also interested in whether $0 \in K$ if $B_i$ are not necessarily subsets of a boundary of some set $C$, rather:

  • they are closed subsets of a boundary of possibly different convex sets $C_i$ (in fact, we can assume that $C_i$ are closed balls in $V$),
  • if $B_i \in B$, it follows that $- B_i \in B$.
  • There exists a Hamel basis $P$ of $V$ such that $p \in P \implies \exists i \in I \quad p \in B_i.$
  • $\forall i \in I \quad 0 \notin B_i$.

Are there any references that could help to answer these questions?


Unfortunately, these properties do not imply that $0 \in K$ even if $\dim V = 3$. The reason being that we can choose a vertex of such a cube, then take 3 vertices such that there exist an edge with that vertex and the chosen one. The convex hull of these 4 vertices do not contain $0$, but it does satisfy the necessary conditions.

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Without loss of generality, $K$ is a polytope with its vertices in the $B_i$'s. By Farkas' lemma, it's equivalent for the vertices to reside on the same side of a subspace of codimension $1$. With this insight, it's easy to find counterexamples. For example, in three dimensions, we can take $K$ to be the convex hull of $(1,1,1)$, $(-1,1,1)$, $(1,-1,1)$, and $(1,1,-1)$:

enter image description here

(The point in the middle is the origin.)

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