Abstract
In this chapter, we study convex functions, which are the analytic counterpart of convex sets. There are many relations between convex functions and sets. Our definition of a convex function via the convexity of its epigraph emphasizes this connection. For convex functions, the study of regularity properties such as continuity or differentiability is particularly natural, but in turn this suggests to consider smoothness of convex sets as well. A strong and very useful link between sets and functions is provided by the support function of a convex set. This tool will be crucial throughout the book.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press, Cambridge (1996)
Hörmander, L.: Notions of Convexity. Birkhäuser, Basel (1994)
Koldobsky, A.: Fourier Analysis in Convex Geometry. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2005)
Koldobsky, A., Yaskin, V.: The Interface Between Convex Geometry and Harmonic Analysis. CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence (2008)
Marti, J.T.: Konvexe Analysis. Birkhäuser, Basel (1977)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Simon, B.: Convexity: An Analytic Viewpoint. Cambridge Tracts in Mathematics, vol. 187. Cambridge University Press, Cambridge (2011)
Stoer, J., Witzgall, Ch.: Convexity and Optimization in Finite Dimensions I. Springer, Berlin (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hug, D., Weil, W. (2020). Convex Functions. In: Lectures on Convex Geometry. Graduate Texts in Mathematics, vol 286. Springer, Cham. https://doi.org/10.1007/978-3-030-50180-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-50180-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-50179-2
Online ISBN: 978-3-030-50180-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)