Self-concordant barriers play a key role in interior-point algorithms for conic programming. However, the optimal barrier parameter is not known for most non-symmetric cones relevant for optimization, like the 3-dimensional power cone and the cone over the unit ball of the p-norms. In [6], the problem of optimizing the barrier parameter for interior point methods was approximated by a semi-definite program. Unfortunately, the self-concordance condition is non-convex, so we have to consider its convex relaxation. The existence of a self-concordant barrier with parameter \(\nu \) can be reduced to the inclusion at each point of the cone of a certain vector in a three-dimensional body \(P_{\nu }\) . The non-convexity of this body is the source of the non-convexity of the original problem. In this paper, we give an exact description of the convex hull of the body and propose an exact function \(\tilde{\nu }(\nu )\) , such that the body \(P_{\tilde{\nu }(\nu )}\) contains the convex hull of the body \(P_{\nu }\) . Thus, by solving the convex semi-definite problem on the existence of a barrier with parameter \(\nu \) , we can obtain a barrier with parameter not worse than \(\tilde{\nu }(\nu )\) .