Beale (1955), Theorem 3: given x, C is a convex function of (A, β)

Fix constants $c\in\mathbb R^n$, $f\in\mathbb R^p$, $D\in\mathbb R^{m\times p}$ and a non-negative first-stage vector $x\in\mathbb R^n$, $x\ge 0$. For data $(A,\beta)\in\mathbb R^{m\times n}\times\mathbb R^m$ let

Let $K$ be a convex set of data pairs $(A,\beta)$ such that for every $(A,\beta)\in K$ the minimum over $y$ is attained. Then $C$ is jointly convex on $K$: for $(A_1,\beta_1),(A_2,\beta_2)\in K$ and $\lambda_1,\lambda_2\ge0$ with $\lambda_1+\lambda_2=1$,

This is Theorem 3 of the paper, "Given $x$, $C$ is a convex function of $A$ and $\beta$". Beale uses it to argue that adding a zero-mean random perturbation to the coefficients cannot decrease $E(C)$.

Formalization Note The paper names no domain for $(A,\beta)$. The statement is made on every convex set of data on which the second-stage minimum is attained, i.e. every convex set on which $C$ is defined as a minimum; attainment is needed because the real infimum is a junk $0$ elsewhere. The first-stage vector $x$ is taken non-negative, as in the paper's model ("choose non-negative $x_j$", p. 181); the page says only "given $x$".