Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes

In this paper we show that the convex envelope over polytopes for a class of bivariate functions, including the bilinear and fractional functions as special cases, is characterized by a polyhedral subdivision of the polytopes, and is such that over each member of the subdivision the convex envelope has a given (although possibly only implicitly defined) functional form. For the bilinear and fractional case we show that there are three possible functional forms, which can be explicitly defined.

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