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The Legendre–Fenchel conjugate of the product of two positive-definite quadratic forms was posted as an open question in the field of nonlinear analysis and optimization by Hiriart-Urruty [‘Question 11’ in SIAM Review 49, 255–273, 2007]. Under a convexity assumption on the function, it was answered by Zhao [SIAM J. Matrix Analysis & Applications 31(4), 1792–1811, 2010]. In this note, we answer the open question without making the convexity assumption.

A Note on Legendre–Fenchel Conjugate of the Product of Two Positive-Definite Quadratic Forms