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Assume that $X_1,\ldots,X_n$ are open, convex subsets of $\Bbb R^d$ such that for any $i,j,k$ with $1\le i,j,k\le n$, we have $X_i\cap X_j\cap X_k\neq\emptyset$. Is it possible for $\bigcap_{i=1}^n X_i$ to be empty?

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For $d\leqslant 2$, the intersection is non-empty by Helly's theorem. If $d>2$, one can denote by $X_1,\ldots,X_{d+1}$ the neighborhoods of facets of a simplex in $\mathbb{R}^d$ to get a counterexample.

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