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Journal of Convex Analysis 30 (2023), No. 1, 143--157
Copyright Heldermann Verlag 2023

On the Roots of Convex Functions

Daniel Solow
Department of Operations, Weatherhead School of Management, Case Western Reserve University, Cleveland, OH 44106, U.S.A.

Fangqi Fu
Magnolia Drive, Cleveland, Ohio, U.S.A.

The number of real roots of a convex function of one variable is shown to be 0, 1, 2, or infinite and, in the latter case, the infinite number of roots must be an interval on the real line. Necessary and sufficient conditions are provided on the convex function for each of these cases to arise. For example, a necessary and sufficient condition for there to be an infinite interval of roots is for the function to have three distinct roots. An algorithm is also presented for finding all of the roots, subject to computational limitations. At the end of the paper, two generalizations are suggested for future directions of research and an open conjecture is presented of a sufficient condition for a system of n convex equations in n unknowns to have a solution.

Keywords: Solving convex equations, convex functions, finding roots, roots of convex functions.

MSC: 52A41.

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