Some 0/1 polytopes need exponential size extended formulations

We prove that there are 0/1 polytopes $${P \subseteq \mathbb{R}^{n}}$$ that do not admit a compact LP formulation. More precisely we show that for every n there is a set $${X \subseteq \{ 0,1\}^n}$$ such that conv(X) must have extension complexity at least $${2^{n/2\cdot(1-o(1))}}$$ . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on $${\mathbf{NP}\not\subseteq \mathbf{P_{/poly}}}$$ , our result rules out the existence of a compact formulation for any $${\mathbf{NP}}$$ -hard optimization problem even if the formulation may contain arbitrary real numbers.