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.

polar conenot dual-cone)....this sounds like a textbook result. $\endgroup$