2000 character limit reached
Quadruple Inequalities: Between Cauchy-Schwarz and Triangle (2307.01361v4)
Published 3 Jul 2023 in math.MG
Abstract: We prove a set of inequalities that interpolate the Cauchy-Schwarz inequality and the triangle inequality. Every nondecreasing, convex function with a concave derivative induces such an inequality. They hold in any metric space that satisfies a metric version of the Cauchy-Schwarz inequality, including all CAT(0) spaces and, in particular, all Euclidean spaces. Because these inequalities establish relations between the six distances of four points, we call them quadruple inequalities. In this context, we introduce the quadruple constant - a real number that quantifies the distortion of the Cauchy-Schwarz inequality by a given function. Additionally, for inner product spaces, we prove an alternative, more symmetric version of the quadruple inequalities, which generalizes the parallelogram law.
- “The best constant in the Topchii-Vatutin inequality for martingales” In Statist. Probab. Lett. 65.3, 2003, pp. 199–206 DOI: 10.1016/j.spl.2003.06.002
- S.M. Buckley, K. Falk and D.J. Wraith “Ptolemaic spaces and CAT(0)” In Glasg. Math. J. 51.2, 2009, pp. 301–314 DOI: 10.1017/S0017089509004984
- Martin R. Bridson and André Haefliger “Metric spaces of non-positive curvature” 319, Grundlehren der mathematischen Wissenschaften Springer-Verlag, Berlin, 1999, pp. xxii+643 DOI: 10.1007/978-3-662-12494-9
- Leonard M. Blumenthal “Theory and applications of distance geometry.” Chelsea Publishing Co., New York,, 1970, pp. xi+347
- “Quasilinearization and curvature of Aleksandrov spaces” In Geom. Dedicata 133, 2008, pp. 195–218 DOI: 10.1007/s10711-008-9243-3
- “Two deterministic half-quadratic regularization algorithms for computed imaging” In Proceedings of 1st International Conference on Image Processing 2 Los Alamitos, CA, USA: IEEE Computer Society, 1994, pp. 168\bibrangessep169\bibrangessep170\bibrangessep171\bibrangessep172 DOI: 10.1109/ICIP.1994.413553
- Paul Corazza “Introduction to metric-preserving functions” In Amer. Math. Monthly 106.4, 1999, pp. 309–323 DOI: 10.2307/2589554
- Michel Marie Deza and Elena Deza “Encyclopedia of distances” Springer, Berlin, 2016, pp. xxii+756 DOI: 10.1007/978-3-662-52844-0
- Per Enflo “On the nonexistence of uniform homeomorphisms between Lpsubscript𝐿𝑝L_{p}italic_L start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT-spaces” In Ark. Mat. 8, 1969, pp. 103–105 DOI: 10.1007/BF02589549
- Per Enflo “Uniform structures and square roots in topological groups. II” In Israel J. Math. 8, 1970, pp. 253–272 DOI: 10.1007/bf02771561
- “Roundness properties of ultrametric spaces” In Glasg. Math. J. 56.3, 2014, pp. 519–535 DOI: 10.1017/S0017089513000438
- Thomas Foertsch, Alexander Lytchak and Viktor Schroeder “Nonpositive curvature and the Ptolemy inequality” In Int. Math. Res. Not. IMRN, 2007, pp. Art. ID rnm100\bibrangessep15 DOI: 10.1093/imrn/rnm100
- Maurice Fréchet “Les éléments aléatoires de nature quelconque dans un espace distancié” In Ann. Inst. H. Poincaré 10, 1948, pp. 215–310
- P.Thomas Fletcher, Suresh Venkatasubramanian and Sarang Joshi “The geometric median on Riemannian manifolds with application to robust atlas estimation” Mathematics in Brain Imaging In NeuroImage 45.1, Supplement 1, 2009, pp. S143–S152 DOI: 10.1016/j.neuroimage.2008.10.052
- Peter J. Green “Bayesian reconstructions from emission tomography data using a modified EM algorithm.” In IEEE transactions on medical imaging 9 1, 1990, pp. 84–93
- Peter J. Huber “Robust estimation of a location parameter” In Ann. Math. Statist. 35, 1964, pp. 73–101 DOI: 10.1214/aoms/1177703732
- Jovan Karamata “Sur une inégalité rélative aux fonctions convexes” In Publ. Math. Univ. Belgrade 1, 1932, pp. 145–148
- Ju.G. Rešetnjak “Non-expansive maps in a space of curvature no greater than K𝐾Kitalic_K.” In Sibirsk. Mat. Ž., 1968, pp. 918–927
- R.Tyrrell Rockafellar “Convex analysis” 28, Princeton Math. Ser. Princeton University Press, Princeton, NJ, 1970, pp. xviii+451 DOI: 10.1515/9781400873173
- Christof Schötz “Convergence rates for the generalized Fréchet mean via the quadruple inequality” In Electron. J. Stat. 13.2, 2019, pp. 4280–4345 DOI: 10.1214/19-EJS1618
- Christof Schötz “Strong laws of large numbers for generalizations of Fréchet mean sets” In Statistics 56.1, 2022, pp. 34–52 DOI: 10.1080/02331888.2022.2032063
- J.Michael Steele “The Cauchy-Schwarz master class” An introduction to the art of mathematical inequalities, AMS/MAA Problem Books Series Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 2004, pp. x+306 DOI: 10.1017/CBO9780511817106
- Karl-Theodor Sturm “Probability measures on metric spaces of nonpositive curvature” In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) 338, Contemp. Math. Amer. Math. Soc., Providence, RI, 2003, pp. 357–390 DOI: 10.1090/conm/338/06080
- “Maximum of the critical Galton-Watson processes and left-continuous random walks” In Teor. Veroyatn. Primen. 42.1, 1997, pp. 21–34 DOI: 10.1137/s0040585x97975903
Sponsor
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.