  The fair allocation of mixed goods, consisting of both divisible and
indivisible goods, has been a prominent topic of study in economics and
computer science. We define an allocation as fair if its utility vector
minimizes a symmetric strictly convex function. This fairness criterion
includes standard ones such as maximum egalitarian social welfare and maximum
Nash social welfare. We address the problem of minimizing a given symmetric
strictly convex function when agents have binary valuations. If only divisible
goods or only indivisible goods exist, the problem is known to be solvable in
polynomial time. In this paper, firstly, we demonstrate that the problem is
NP-hard even when all indivisible goods are identical. This NP-hardness is
established even for maximizing egalitarian social welfare or Nash social
welfare. Secondly, we provide a polynomial-time algorithm for the problem when
all divisible goods are identical. To accomplish these, we exploit the
proximity structure inherent in the problem. This provides theoretically
important insights into the hybrid domain of convex optimization that
incorporates both discrete and continuous aspects.
