# Minimum volume-covering ellipoid as a semidefinite program (SDP)

Consider the problem of finding a minimal volume-covering ellipsoid:

$$\begin{array}{ll} \text{minimize} & \log \det X^{-1}\\ \text{subject to} & a_i^T X a_i \le 1\\ & X \succeq 0\end{array}$$

I have two questions:

1. Can the problem above be written as a semidefinite program (SDP)?

2. If the answer to question 1 is yes, how to write it to standard form of SDP as follows:

$$minimize \quad tr(CX)$$ $$s.t. \quad tr(A_iX)=b_i, \quad i=1,2,...,p$$ $$X \succeq 0$$ In fact,I confirm that it is a convex programming because $\log \det X^{-1}$ is a convex function and $a_i^TXa_i \le 1 \iff tr(a_ia_i^TX) \le 1$ is linear constraint for $X \in S^n_{++}$.