Borel measures with a density on a compact semi-algebraic set

2013/04/05 by Jean B. Lasserre, Lasserre, Jean-Bernard
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Advanced Topology and Set Theory #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis

paper · doi:10.48550/arxiv.1304.1716

openalex publication_date 2013/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K⊂ Rⁿ be a compact basic semi-algebraic set. We provide a necessary and sufficient condition (with no a priori bounding parameter) for a real sequence y=(y_α), α∈ Nⁿ, to have a finite representing Borel measure absolutely continuous w.r.t. the Lebesgue measure on K, and with a density in ∩_p=1^∞ L_p(K). With an additional condition involving a bounding parameter, the condition is necessary and sufficient for existence of a density in L_∞(K). Moreover, nonexistence of such a density can be detected by solving finitely many of a hierarchy of semidefinite programs. In particular, if the semidefinite program at step d of the hierarchy has no solution then the sequence cannot have a representing measure on K with a density in L_p(K) for any p≥ 2d.

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