The Lipschitz Condition of the Metric Projection in the Pliś Metric

Maxim V. Balashov

doi:10.68381/jca27046

Abstract

We consider the class of Banach spaces where the metric projection on a convex closed bounded set is Lipschitz continuous in the Plis metric with respect to the set. We obtain some geometrical and analytical conditions for this property. In fact, the space should has a Hilbert-like structure.

Author information

Contact details are reproduced from the original publication and may be historical.

Maxim V. Balashov

V. A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow 117997, Russia

balashov73@mail.ru

Keywords

Uniform convexity
uniform smoothness
Banach space
Hilbert space
metric projection
Plis metric
moduli of convexity and smoothness

Mathematics Subject Classification

49J53, 52A07, 46C05, 26B25; 46B25, 46B20

References

M. V. Balashov: The Plis metric and Lipschitz stability of minimization problems, Sbornik: Mathematics 210/7 (2019) 911–927.https://doi.org/10.1070/SM9128 Google Scholar
M. V. Balashov: Lipschitz stability of extremal problems with a strongly convex set, J. Convex Analysis 27/1 (2020) 103–116.https://doi.org/10.68381/jca27007 Google Scholar
M. V. Balashov, D. Repovš: Uniform convexity and the splitting problem for selections, J. Math. Anal. Appl. 360 (2009) 307–316.https://doi.org/10.1016/j.jmaa.2009.06.045 Google Scholar
M. V. Balashov, D. Repovš: Weakly convex sets and modulus of nonconvexity, J. Math. Anal. Appl. 371/1 (2010) 113–127.https://doi.org/10.1016/j.jmaa.2010.04.039 Google Scholar
M. V. Balashov, D. Repovš: On Pliś metric on the space of strictly convex compacta, J. Convex Analysis 19/1 (2012) 171–183.https://doi.org/10.68381/jca19010 Google Scholar
J. Diestel: Geometry of Banach Spaces – Selected Topics, Lecture Notes in Mathematics 485, Springer, Berlin (1975).https://doi.org/10.1007/BFb0082079 Google Scholar
J. Lindenstrauss: On nonlinear projections in Banach spaces, Michigan Math. J. 11/3 (1964) 263–287.https://doi.org/10.1307/mmj/1028999141 Google Scholar
A. Pliś: Uniqueness of optimal trajectories for non-linear control system, Ann. Polom. Math. 29 (1975) 397–401.https://doi.org/10.4064/ap-29-4-397-401 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.

[1] M. V. Balashov: The Plis metric and Lipschitz stability of minimization problems, Sbornik: Mathematics 210/7 (2019) 911–927.https://doi.org/10.1070/SM9128 Google Scholar

[2] M. V. Balashov: Lipschitz stability of extremal problems with a strongly convex set, J. Convex Analysis 27/1 (2020) 103–116.https://doi.org/10.68381/jca27007 Google Scholar

[3] M. V. Balashov, D. Repovš: Uniform convexity and the splitting problem for selections, J. Math. Anal. Appl. 360 (2009) 307–316.https://doi.org/10.1016/j.jmaa.2009.06.045 Google Scholar

[4] M. V. Balashov, D. Repovš: Weakly convex sets and modulus of nonconvexity, J. Math. Anal. Appl. 371/1 (2010) 113–127.https://doi.org/10.1016/j.jmaa.2010.04.039 Google Scholar

[5] M. V. Balashov, D. Repovš: On Pliś metric on the space of strictly convex compacta, J. Convex Analysis 19/1 (2012) 171–183.https://doi.org/10.68381/jca19010 Google Scholar

[6] J. Diestel: Geometry of Banach Spaces – Selected Topics, Lecture Notes in Mathematics 485, Springer, Berlin (1975).https://doi.org/10.1007/BFb0082079 Google Scholar

[7] J. Lindenstrauss: On nonlinear projections in Banach spaces, Michigan Math. J. 11/3 (1964) 263–287.https://doi.org/10.1307/mmj/1028999141 Google Scholar

[8] A. Pliś: Uniqueness of optimal trajectories for non-linear control system, Ann. Polom. Math. 29 (1975) 397–401.https://doi.org/10.4064/ap-29-4-397-401 Google Scholar