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It is proved that every discrete Morse function in the sense of Forman on afinite regular CW complex can be represented by a polyhedral Morse function inthe sense of Banchoff on an appropriate embedding in Euclidean space of thebarycentric subdivision of the CW complex; such a representation preservescritical points. The proof is stated in terms of discrete Morse functions on aclass of posets that is slightly broader than the class of face posets offinite regular CW complexes.

Polyhedral Representation of Discrete Morse Functions on Regular CW Complexes and Posets