Ph.D Student | Igor Vestfrid |
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Subject | Affine Properties of Almost Isometries and of Almost Jensen Functions |

Department | Department of Mathematics |

Supervisor | Professor Emeritus Benyamini Yoav |

Mazur and Ulam proved in 1932 that a surjective isometry between real normed spaces X and Y is necessarily affine, i.e., that the linear structure of the spaces can be recovered from their metric structure. In the 1940's Ulam proposed to study what can be said when the spaces are only “nearly isometric”. Of the several natural notions of “nearly isometric”, Hyers and Ulam introduced in 1945 the notion of an ε-isometry, i.e., a map f: X → Y so that | || f(y)-f(x)|| - ||y-x|| | ≤ ε for every x and y, and proved that every surjective ε-isometry between real Hilbert spaces can be Cε approximated by a surjective isometry. This result was generalized by several authors (including Gruber, Gevirtz and Omladič and Šemrl) to general Banach spaces, and the sharp value of C is now known to be 2.

In the 1960's John studied an essentially local notion of “nearly isometric”, namely, that of a quasi-isometry and obtained, among other results, that ε-quasi-isometries of bounded convex subsets of Euclidean spaces can be approximated by affine isometries.

The present thesis is a continuation of this project.

A basic tool here is the study of approximate solutions of the Jensen functional equation, i.e., of the equation 2f((x+y)/2) - f(x) - f(y) = 0. In the first part of the thesis we introduce and study the properties of a new class of such approximate solutions, the so-called Approximately Midlinear (AML) maps. We approximate these maps by true Jensen functions. Among other things we generalize some results on affine approximation of approximate solutions of Jensen's equation due to Brudnyi and Kalton, Dilworth, Howard and Roberts, Laczkovich and Behrends and Nikodem.

In the second part we show that ε-isometries from bounded sets into a uniformly convex Banach space as well as all ε-quasi-isometries are AML maps. Combining these results with the results of the first part we obtain, in the spirit of Ulam's program, affine approximations of ε-isometries and of ε-quasi-isometries. These affine approximations are also nearly isometries (i.e., small into-isomorphisms), and in certain situations the linear theory of Banach spaces actually gives an approximation of such linear near-isometries by isometries. The results in this part extend previous results of several authors, including John, Gevirtz and Väisälä. We also improve some results of Gevirtz on the injectivity of quasi-isometries.