TY - JOUR

T1 - Polygonal equalities and virtual degeneracy in Lp-spaces

AU - Kelleher, Casey

AU - Miller, Daniel

AU - Osborn, Trenton

AU - Weston, Anthony

N1 - Funding Information:
The research presented in this paper was initiated at the 2011 Cornell University Summer Mathematics Institute (SMI) and completed under the auspices of the Visiting Researcher Programme at the University of South Africa (UNISA). The authors would like to thank the Department of Mathematics and the Center for Applied Mathematics at Cornell University for supporting this project, and the National Science Foundation for its financial support of the SMI through NSF grant DMS-0739338 . We are very grateful for the additional financial support from UNISA , Canisius College and the University of New South Wales that aided in the completion of this paper. In addition, we would like to thank Petrus Potgieter, Willem Fouché, Ian Doust and Stephen Sánchez for their particularly helpful input on (infinitely many) preliminary drafts of this paper. The last named author extends special thanks to the Australian Catholic University for additional support through their programme of Honorary professorships.

PY - 2014/7/1

Y1 - 2014/7/1

N2 - Suppose 0p(Ω, μ) is at least two-dimensional. The central results of this paper provide a complete description of the subsets of Lp(Ω, μ) that have strict p-negative type. In order to do this we study non-trivial p-polygonal equalities in Lp(Ω, μ). These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form∑j,i=1nαjαizj-zipp=0 where {z1,..., zn} is a subset of Lp(Ω, μ) and α1,..., αn are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial p-polygonal equalities in Lp(Ω, μ). The cases p<2 and p=2 are substantially different and are treated separately. The case p=1 generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial p-polygonal equalities in Lp(Ω, μ) is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if (X, d) is a metric space that has strict q-negative type for some q≥p, then: (1) (X, d) is not isometric to any linear subspace W of Lp(Ω, μ) that contains a pair of disjointly supported non-zero vectors, and (2) (X, d) is not isometric to any subset of Lp(Ω, μ) that has non-empty interior. Furthermore, in the case p=2, it also follows that (X, d) is not isometric to any affinely dependent subset of L2(Ω, μ). More generally, we show that if (Y, ρ) is a metric space whose generalized roundness ℘ is finite and if (X, d) is a metric space that has strict q-negative type for some q≥℘, then (X, d) is not isometric to any metric subspace of (Y, ρ) that admits a non-trivial p1-polygonal equality for some p1∈[℘, q]. It is notable in all of these statements that the metric space (X, d) can, for instance, be any ultrametric space. As a result we obtain new insights into sophisticated embedding theorems of Lemin and Shkarin. We conclude the paper by constructing some pathological infinite-dimensional linear subspaces of ℓp that do not have strict p-negative type.

AB - Suppose 0p(Ω, μ) is at least two-dimensional. The central results of this paper provide a complete description of the subsets of Lp(Ω, μ) that have strict p-negative type. In order to do this we study non-trivial p-polygonal equalities in Lp(Ω, μ). These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form∑j,i=1nαjαizj-zipp=0 where {z1,..., zn} is a subset of Lp(Ω, μ) and α1,..., αn are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial p-polygonal equalities in Lp(Ω, μ). The cases p<2 and p=2 are substantially different and are treated separately. The case p=1 generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial p-polygonal equalities in Lp(Ω, μ) is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if (X, d) is a metric space that has strict q-negative type for some q≥p, then: (1) (X, d) is not isometric to any linear subspace W of Lp(Ω, μ) that contains a pair of disjointly supported non-zero vectors, and (2) (X, d) is not isometric to any subset of Lp(Ω, μ) that has non-empty interior. Furthermore, in the case p=2, it also follows that (X, d) is not isometric to any affinely dependent subset of L2(Ω, μ). More generally, we show that if (Y, ρ) is a metric space whose generalized roundness ℘ is finite and if (X, d) is a metric space that has strict q-negative type for some q≥℘, then (X, d) is not isometric to any metric subspace of (Y, ρ) that admits a non-trivial p1-polygonal equality for some p1∈[℘, q]. It is notable in all of these statements that the metric space (X, d) can, for instance, be any ultrametric space. As a result we obtain new insights into sophisticated embedding theorems of Lemin and Shkarin. We conclude the paper by constructing some pathological infinite-dimensional linear subspaces of ℓp that do not have strict p-negative type.

KW - Generalized roundness

KW - Isometry

KW - Polygonal equality

KW - Strict negative type

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U2 - 10.1016/j.jmaa.2014.01.063

DO - 10.1016/j.jmaa.2014.01.063

M3 - Article

AN - SCOPUS:84897602564

VL - 415

SP - 247

EP - 268

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -