 Solution

For $v\in S^{n-1}$, orthogonally project every member of ${\mathcal{C}}_i$ onto the oriented line ${\ell }_v$. These projections are compact intervals. They are pairwise intersecting because the original sets are, and pairwise intersecting intervals have a common intersection. Let $m_i(v)$ be the midpoint of that common interval and let $f_i(v)$ be its signed coordinate on ${\ell }_v$. Compactness makes $f_i$ continuous, and reversing the orientation gives

Define the continuous antipodal mapThe Borsuk-Ulam theorem gives $v$ with $F(v)=0$. Write the common value as $t$. The hyperplanemeets every set in every ${\mathcal{C}}_i$, because $t=f_i(v)$ lies in the projection interval of each such set. Thus $H$ is the required common hyperplane transversal.