Suppose that $P$ is a compact and convex polytope in $R^d$ and let $G$ be the graph of $P$ ($V(G)$ are the vertices of $P$ and $E(G)$ are the $1$-dimensional faces - for example polyedral graphs are the graphs of polytopes in $R^3$ http://en.wikipedia.org/wiki/Polyhedral_graph)
By Balinski's theorem (http://en.wikipedia.org/wiki/Balinski%27s_theorem) we know that those graphs are $d$-vertex-connected (http://en.wikipedia.org/wiki/Connectivity_%28graph_theory%29).
But I don't know how to prove something stronger. Suppose that we are given a hyperplane $L$ which splits $P$ into two parts. Let $x$, $y$ be two vertices located in the same part of $P$. I would like to prove that there exists a path in $G$, between $x$ and $y$, such that the path uses only vertices from the same part of $P$.