Suppose that I have a strongly convex function $f(\mathbf{x}): \mathbb{R}^m \rightarrow \mathbb{R}$. Is the Legendre transform of this function also strongly convex?
As far as I can tell, strict convexity of $f$ implies strict convexity of the Legendre transform of $f$ (as shown in https://proofwiki.org/wiki/Convexity_of_Function_implies_Convexity_of_its_Legendre_Transform). However, I am unsure as to whether the same argument holds for strongly convex $f$.
EDIT: The following link appears to support this claim for the case of $m =1$. https://books.google.ca/books?id=HjznBwAAQBAJ&pg=PA86&lpg=PA86&dq=linear+programming+vanderbei+strongly+convex+legendre+transform&source=bl&ots=mBLOVV7_a0&sig=xMI0kBz7bHdN4kkEnvQQ87LGkhk&hl=en&sa=X&ved=0ahUKEwiayNCfupXVAhVFeSYKHQI_Dl8Q6AEIJjAA#v=onepage&q=linear%20programming%20vanderbei%20strongly%20convex%20legendre%20transform&f=false