On The Measure Of Two-Dimensional Regions With Polynomial-Time Computable Boundaries

Authors:
Ker-I Ko;K. Weihrauch
Affiliations:
-;-
Venue:
CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
Year:
1996

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Abstract

We study the computability of the Lebesgue measure of a two-dimensional region that has a polynomial-time computable boundary. It is shown that the two-dimensional measure of the boundary itself completely characterizes the computability of the measure of the interior region. Namely, if a polynomial-time computable, simple, closed curve has measure zero, then its interior region must have a computable measure. Conversely, if such a curve has a positive measure, then the measure of its interior region could be any positive, left r.e. real number.