Consider a function $f: \mathbb{R}^4 \to \mathbb{R}$ such that
$$f(x_1, x_2, y_1, y_2) = p x_1 (x_1 - y_1) + (1 - p) x_2 (x_2 - y_2)$$
for some $p \in (0, 1)$. I'd like to check whether the set
$$C = \{ z \in \text{(feasible area)} \subseteq \mathbb{R}^4 \mid f(z) \leq 0 \}$$
is convex or not in the following feasible area: for both $i = 1, 2$, \begin{align} x_i + y_i &= 1 \land x_i \geq 0 \land y_i \geq 0. \end{align}
If I first substitute two equality constraints into the original objective function $f$, then $f$ is convex w.r.t. $(x_1, x_2)$, which implies $C$ is convex.
Then is there a way to check the convexity of $C$, without reducing the number of variables first?