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\begin{document}$${\mathcal{C}}$$\end{document} be the convex hull of points \documentclass[12pt]{minimal}
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\begin{document}$${{\{{1 \choose x}{1 \choose x}^T \,|\, x\in \mathcal{F}\subset \Re^n\}}}$$\end{document}. Representing or approximating \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{C}}$$\end{document} is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables. We show that if n ≤ 4 and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}}$$\end{document} is a simplex, then \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{C}}$$\end{document} has a computable representation in terms of matrices X that are doubly nonnegative (positive semidefinite and componentwise nonnegative). We also prove that if n = 2 and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}}$$\end{document} is a box, then \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{C}}$$\end{document} has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation-linearization technique (RLT). The simplex result generalizes known representations for the convex hull of \documentclass[12pt]{minimal}
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\begin{document}$${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$\end{document} when \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}\subset\Re^2}$$\end{document} is a triangle, while the result for box constraints generalizes the well-known fact that in this case the RLT constraints generate the convex hull of \documentclass[12pt]{minimal}
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\begin{document}$${{\{(x_1, x_2, x_1x_2)\,|\, x\in\mathcal{F}\}}}$$\end{document}. When n = 3 and \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{F}}$$\end{document} is a box, we show that a representation for \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{C}}$$\end{document} can be obtained by utilizing the simplex result for n = 4 in conjunction with a triangulation of the 3-cube.
