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This study addresses sets of lower bounds in a vector space ordered by a convex cone. It is\udeasy to see that every set of lower bounds is downward (lower?), bounded from above, with the further\udproperty that it contains the supremum of any of its subsets which admits one. Our main result proves that\udthese conditions are also sufficient, if the ordering cone is polyhedral. We provide other characterizations\udand properties of sets of lower bounds in primal and dual terms and show by means of simple counterexamples\udthat such results fail when the polyhedrality assumption is dropped

Characterizing Sets of Lower Bounds: a Hidden Convexity Result