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Square distance functions are Polyak-{\L}ojasiewicz and vice-versa

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arxiv 2301.10332 v1 pith:OIHNXTYM submitted 2023-01-24 math.OC

classification math.OC
keywords distancefunctionojasiewiczpolyak-squaredbelowboundedclosed
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This short note gathers known results to state that the squared distance function to a (nonconvex) closed set of an Euclidean space is Polyak-{\L}ojasiewicz. As a fuzzy reciprocate, we also recall that every Polyak-{\L}ojasiewicz function can be bounded from below by the squared distance function to its set of minimizers.

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Cited by 1 Pith paper

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  1. On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences

    math.OC 2025-07 conditional novelty 8.0 of 10

    The convergence rate of the moment-SOS hierarchy on a compact semi-algebraic set is O(1/r^L), where L is the Lojasiewicz exponent of the set's defining polynomials.

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