We have $n+1$ linear matrix inequalities (LMIs) in $\mathrm X$, namely,

$$\mathrm X \succeq \mathrm O_m, \qquad \mathrm P_1 - \mathrm X \succeq \mathrm O_m, \qquad \mathrm P_2 - \mathrm X \succeq \mathrm O_m, \qquad \cdots \qquad \mathrm P_n - \mathrm X \succeq \mathrm O_m$$

The conjunction of these $n+1$ LMIs can be written as a single LMI

$$\begin{bmatrix} \mathrm X & & & & \\ & \mathrm P_1 - \mathrm X & & & \\ & & \mathrm P_2 - \mathrm X & & \\ & & & \ddots & \\ & & & & \mathrm P_n - \mathrm X\end{bmatrix} \succeq \mathrm O_{m (n+1)}$$

which defines a (convex) spectrahedron in $\mathbb R^{\binom{m+1}{2}}$. In the interior of this spectrahedron, the block matrix above is positive definite. At the boundary, the block matrix is merely positive *semidefinite*, i.e., its rank is lower at the boundary.

From Sylvester's criterion for positive semidefiniteness, saying that $\mathrm P_i - \mathrm X \succeq \mathrm O_m$ is equivalent to saying that all $2^m - 1$ principal minors of matrix $\mathrm P_i -\mathrm X$ are nonnegative. In other words, each LMI of the form $\mathrm P_i - \mathrm X \succeq \mathrm O_m$ encapsulates at most $2^m - 1$ polynomial inequalities in the $\binom{m+1}{2}$ entries of symmetric $\rm X$. Thus, in total, we have at most $(2^m - 1) (n+1)$ polynomial inequalities.