I've been busting my brains about proving the following theorem:
Given that $A,B,C\in S^n$ with $n\geq3$, and that for some $\alpha, \beta,\gamma\in\mathbb{R}$ it holds that $\alpha A+\beta B+\gamma C\succcurlyeq 0$, show the set $K=\{X\in S^n_+ \mid \langle A,X \rangle =a, \langle B,X\rangle =b, \langle C,X \rangle=c\}$ is bounded.
Where ofcourse $\langle \cdot, \cdot \rangle$ is the matrix inner product $\langle X,Y \rangle = \sum\limits_{i,j=1}^n X_{i,j}Y_{i,j}$
Theorem is false. Consider $$A = B = C = \begin{bmatrix} 1 & 0 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \text{ and } a = b =c = 1. $$ Then $A+B+C \succeq 0$, but the set $K$ is not bounded. For example, $$X_M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & M & 0 \\ 0 & 0 & 0 \end{bmatrix} \in K \quad \forall M \geq 0,$$ and clearly, $$ \lim_{M \to \infty} \| X_M\| \to \infty.$$