I wonder if there is any existing result for self-concordant function in the literature about the following question. 


Suppose $f$ is a self-concordant barrier function of a proper cone $K$ (pointed, closed,convex and solid), is there any result for a self-concordant barrier function $g$ of the dual of $K$, i.e. $K^\*$? Is there a formula to get $g$ from $f$ or any relation between the two?



Self-concordant function for dual cone
It is true that the Fenchel dual of a self-concordant function $f$ on a cone $K$ is also self-concordant (with the same parameter) and the domain of $f^\*$ will be the image of the gradient of $f$, i.e. the natural domain of $f^\*$. This happens to coincide with $K^\*$ if $f$ is logarithmically homogeneous or if $f$ is minus the logarithm of a hyperbolic polynomial(the second is only different from the first if the polynomial is not homogeneous), but I don't think it is true in general: if for example $f$ is coercive, the image of the gradient will include non zero points of $K$. 


Some authors don't include this but I'm assuming that the definition of self-concordant includes that the interior of $K$ is the natural domain of $f$ (i.e. f goes to infinity in the boundary of $K$). Also, I'm using $K^\* = \{ y | \forall x \in K: y^Tx \leq 0\}$, which I guess wikipedia calls the polar cone, but the difference is only a minus sign so it shouldn't be a problem.


