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 {\it SIAM Review} 49, 255-273, (2007)]. Under a convex 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.
Citation
Yong Xia, A note on Legendre-Fenchel conjugate of the product of two positive-definite quadratic forms, Journal of Operations Research Society of China, to appear, 2013