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.