Competitive equilibrium in Leontief economies Consider an economy in which all consumers have, possibly different, Leontief utilities. Since preferences are not strictly convex, it is not guaranteed that a competitive equilibrium exists. I found some papers that discuss the computational problem of deciding whether a Leontief economy has a competitive equilibrium, but I am interested in general existence results:

A. What conditions on Leontief economies guarantee that a competitive equilibrium exists?

B. In particular, if the initial endowments are equal (each of $m$ agents receives a fraction $1/m$ of each good), is a competitive equilibrium guaranteed to exist?
Strict convexity of preferences is not needed in existence results for competitive equilibria. Leontief preferences are quite well-behaved. They are continuous, convex, and strongly monotonic.   If all endowments are strictly positive, the existence of a competitive equilibrium in an exchange economy (or a production economy satisfying standard conditions) exists by the first result of the original Arrow-Debreu paper.

Arrow-Debreu actually do not just require convexity, they make, as pointed out by denesp in a comment, the convexity assumption (III.c) on utility functions that $u(x)&gt;u(x')$ and $0&lt;t&lt;1$ implies $u(tx+(1-t)x')&gt;u(x')$. Plain convexity suffices for existence, but Leontief preferences do also satisfy condition (III.c).:  Assume $\min\{\alpha_i x_i\}&gt;\min\{\alpha_i x_i'\}$. Then
$$\min\big\{\alpha_i (tx_i+(1-t)x_i')\big\}&gt;\min\big\{\alpha_i tx_i\big\}+\min\big\{\alpha_i(1-t) x_i'\big\}$$ $$=t\min\{\alpha_i x_i\}+(1-t)\min\{\alpha_i x_i'\}&gt;\min\{\alpha_i x_i'\}.$$