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## Main Question or Discussion Point

I have been trying to figure this out for a couple weeks now. Why does the Legendre transform require that the function be convex?

Is it because g(x) has to be solved to get x(g) and finding this inverse means g(x) should be bijective? (And if g is bijective then dg/dx will always be positive or always be negative [or perhaps one spot be zero] which means always convex (or always concave).)

Is it because g(x) has to be solved to get x(g) and finding this inverse means g(x) should be bijective? (And if g is bijective then dg/dx will always be positive or always be negative [or perhaps one spot be zero] which means always convex (or always concave).)