While reading a paper *An Arithmetic Proof of John’s Ellipsoid Theorem* by Gruber and Schuster, I have a question on their proof.

Consider an $n\times n$ real symmetric and positive definite matrix $\mathbf A$.

- As this kind of matrix is symmetric, its $n(n+1)/2$ upper diagonal terms are enough to represent it. Hence, we can consider such a matrix as a point in $\mathbb R^{n(n+1)/2}$.
- A conical combination of two positive definite matrices is also positive definite. Hence, the set of all symmetric positive definite matrices forms an open convex cone $\mathcal P\in\mathbb R^{n(n+1)/2}$ with apex on the origin.

Now they claim the following theorem without proving it.

The set $\ \mathcal D = \{\mathbf A \in \mathcal P: \det \mathbf A \geq 1\}$ is a closed, smooth, strictly convex set in $\mathcal P$ with non-empty interior.Theorem:

They just gave some hint that we can use Implicit Function Theorem and Minkowski's Determinant Inequality which states that

For two $n\times n$ positive semidefinite Hermitian matrices $\mathbf X$ and $\mathbf Y$, $$\det (\mathbf X + \mathbf Y)^{1/n}\geq \det(\mathbf X)^{1/n} + \det(\mathbf Y)^{1/n} $$

Any hint or suggestion on how to prove the above theorem about the set $\mathcal D$?