Abstract
We investigate computability of convex sets restricted to rational inputs. Several quite different algorithmic characterizations are presented, like the existence of effective approximations by polygons or effective line intersection tests. We also consider approximate computations of the n-fold characteristic function for several natural classes of convex sets. This yields many different concrete examples of (1, n)-computable sets.
Preview
Unable to display preview. Download preview PDF.
References
Manindra Agrawal, V. Arvind. Geometric sets of low information content. School of Mathematics, SPIC Science Foundation, Internal Report TCS-94-5, Madras, June 1994.
Patrick Assouad. Densité et dimension. Ann. Inst. Fourier, Grenoble, 33, 3:233–282, 1983.
R. Beigel, W. I. Gasarch, J. Gill, J. C. Owings, Jr. Terse, superterse, and verbose sets. Information and Computation, 103:68–85, 1993.
Richard Beigel, Martin Kummer, Frank Stephan. Approximable sets. In: Proceedings Structure in Complexity Theory. Ninth Annual Conference, pp. 12–23, IEEE Computer Society Press, 1994.
Anselm Blumer, Andzej Ehrenfeucht, David Haussler, Manfred K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. Journal of the Association for Computing Machinery, 36, 4: 929–965, 1989.
Arne Brøndsted. An introduction to convex polytopes. Springer-Verlag, New York, 1983.
Richard Mansfield Dudley. A course on empirical processes. In: École d'Été de Probabilités de Saint-Flour XII — 1982. Lecture Notes in Mathematics 1097, Springer-Verlag, Berlin, 2–142, 1984.
William I. Gasarch. Bounded queries in recursion theory: a survey. In: Proceedings Structure in Complexity Theory. Sixth Annual Conference, pp. 62–78, IEEE Computer Society Press, 1991.
Xiaolin Ge, Anil Nerode. On extreme points of convex compact Turing located sets. In: Proceedings LFCS'94, St. Petersburg, Russia, July 1994, pp. 114–128, Lecture Notes in Computer Science Vol. 813, Springer-Verlag, Berlin, 1994.
Martin Grötschel, László Lovász, Alexander Schrijver. Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, Heidelberg, 1988.
David Haussler, Emo Welzl. ε-nets and simplex range queries. Discrete Comput. Geometry, 2:127–151, 1987.
Konrad Jacobs. Extremalpunkte konvexer Mengen. In: Selecta Mathematica III, Springer-Verlag, Heidelberg, 1971.
Ker-I Ko. Complexity theory of real functions. Birkhäuser, Boston, 1991.
Jürg T. Marti. Konvexe Analysis. Birkhäuser, Basel, 1977.
H. Minkowski. Gesammelte Abhandlungen, Bd. II. Teubner, Leipzig, 157–161, 1911.
Marian Boykan Pour-El, Jonathan Ian Richards. Computability in analysis and physics. Springer-Verlag, Berlin, 1989.
Marcus Schäfer. Anfragekomplexität geometrischer Mengen. Diplomarbeit, Fakultät für Informatik, Universität Karlsruhe, January 1994.
V. N. Vapnik, A. Ya. Červonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl., 16:264–280, 1971.
R. S. Wencour, R. M. Dudley. Some special Vapnik-Chervonenkis classes. Discrete Mathematics, 33:313–318, 1981.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kummer, M., Schäfer, M. (1995). Computability of convex sets. In: Mayr, E.W., Puech, C. (eds) STACS 95. STACS 1995. Lecture Notes in Computer Science, vol 900. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59042-0_104
Download citation
DOI: https://doi.org/10.1007/3-540-59042-0_104
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59042-2
Online ISBN: 978-3-540-49175-0
eBook Packages: Springer Book Archive