A dichotomy of sets via typical differentiability

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A dichotomy of sets via typical differentiability. / Dymond, Michael; Maleva, Olga.

In: Forum of Mathematics, Sigma, Vol. 8, e41, 04.11.2020.

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@article{490485dce978499b8f6f27d11f08fa7f,
title = "A dichotomy of sets via typical differentiability",
abstract = "We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has zero length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: In any such coverable set a typical Lipschitz function is everywhere severely non-differentiable.",
keywords = "differentiability of Lipschitz functions, Baire category, purely unrectifiable, Banach-Mazur game",
author = "Michael Dymond and Olga Maleva",
year = "2020",
month = nov,
day = "4",
doi = "10.1017/fms.2020.45",
language = "English",
volume = "8",
journal = "Forum of Mathematics, Sigma",
issn = "2050-5094",
publisher = "Cambridge University Press",

}

RIS

TY - JOUR

T1 - A dichotomy of sets via typical differentiability

AU - Dymond, Michael

AU - Maleva, Olga

PY - 2020/11/4

Y1 - 2020/11/4

N2 - We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has zero length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: In any such coverable set a typical Lipschitz function is everywhere severely non-differentiable.

AB - We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function, namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has zero length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: In any such coverable set a typical Lipschitz function is everywhere severely non-differentiable.

KW - differentiability of Lipschitz functions

KW - Baire category

KW - purely unrectifiable

KW - Banach-Mazur game

UR - https://www.cambridge.org/core/journals/forum-of-mathematics-sigma

U2 - 10.1017/fms.2020.45

DO - 10.1017/fms.2020.45

M3 - Article

VL - 8

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

SN - 2050-5094

M1 - e41

ER -