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arXiv · 2101.03755

Scaling-invariant functions versus positively homogeneous functions

Cheikh Touré·Armand Gissler·Anne Auger·Nikolaus Hansen
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Abstract

Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are scaling-invariant with respect to zero. We prove in this paper that the reverse is true for large classes of scaling-invariant functions. Specifically, we give necessary and sufficient conditions for scaling-invariant functions to be composites of a strictly monotonic function with a positively homogeneous function. We also study sublevel sets of scaling-invariant functions generalizing well-known properties of positively homogeneous functions.

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Cheikh Touré, Armand Gissler, Anne Auger, Nikolaus Hansen. 2021-01-11. Scaling-invariant functions versus positively homogeneous functions. https://arxiv.org/abs/2101.03755

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