= 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
$$
f_i(-v)=-f_i(v).
$$
Define the continuous antipodal map
$$
F:S^{n-1}\longrightarrow\mathbb R^{n-1},
\qquad
F(v)=(f_1(v)-f_n(v),\ldots,f_{n-1}(v)-f_n(v)).
$$
The <Borsuk-Ulam theorem> gives $v$ with $F(v)=0$. Write the common value as $t$. The hyperplane
$$
H=\{x\in\mathbb R^n:x\cdot v=t\}
$$
meets 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.
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