Abstract
In this chapter, we probe several topics that use significant ideas from convexity theory and that have significant applications in various fields. In particular, we prove theorems of Radon, Helly, Kirchberger, Bárány, and Tverberg on the combinatorial structure of convex sets, application of Helly’s theorem to semi-infinite programming, in particular to Chebyshev’s approximation problem, homogeneous convex functions, and their applications to inequalities, attainment of optima in maximization of convex functions, decompositions of convex cones, and finally the relationship between the norms of a homogeneous polynomial and its associated symmetric form. The last result has an immediate application to self-concordant functions in interior-point algorithms.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer New York
About this chapter
Cite this chapter
Güler, O. (2010). Topics in Convexity. In: Foundations of Optimization. Graduate Texts in Mathematics, vol 258. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68407-9_13
Download citation
DOI: https://doi.org/10.1007/978-0-387-68407-9_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-34431-7
Online ISBN: 978-0-387-68407-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)