Theorem 7, p. 29 — if the quadratic eigenvalue multiplicity satisfies k ≥ m + 2, then conv(𝒟) = 𝒟_SDP

Consider a QCQP (1) in $N$ variables with $m\ge1$ constraints, $q_i(x) = x^\top A_ix + 2b_i^\top x + c_i$, its epigraph $\mathcal D$ (2) and the projected epigraph $\mathcal D_{\mathrm{SDP}}$ (4) of its Shor SDP relaxation. Suppose Assumption 1 holds: the QCQP is feasible and some $\gamma^*\in\mathbb R^m$ with $\gamma^*_i\ge 0$ for $i\in[\![m_I]\!]$ has $A_0+\sum_i\gamma^*_iA_i\succ0$. If the quadratic eigenvalue multiplicity $k$ (the largest $k$ with $A_i = I_k\otimes\mathbb A_i$ for all $i\in[\![0,m]\!]$, Definition 3) satisfies $k\ge m+2$, then

Unlike Theorems 1 and 2 of the paper, this result needs neither the polyhedrality of $\Gamma$ (Assumption 3) nor the attainment Assumption 2: enough block symmetry in the quadratic forms alone forces the Shor relaxation to describe the convex hull of the epigraph exactly.

Formalization Note Constraint indices are shifted to $\{0,\dots,m-1\}$, with the first $m_I$ being inequalities. $\operatorname{conv}$ is Mathlib's convexHull in $\mathbb R^N\times\mathbb R$. The multiplicity enters as IsQuadEigMult k (the largest admissible $k$), exactly as on the page; for $N = 0$ no such $k$ exists and the statement is vacuous, a degenerate case the paper does not consider.