Uniformly convex functions on Banach spaces

By Jonathan Michael Borwein, A. Guirao, P. Hájek, J. Vanderwerff. 2008

49 citations recorded by the source corpus.

We study the connection between uniformly convex functions f : X -> R bounded above by ||x||^p, and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f : X -> R bounded above by ||.||2 if and only if X admits a norm with modulus of convexity of power type 2.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0dfc7ad0a98042cc17839190ccbdf85e17a437c9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. J. Vanderwerff
2. Michael I. Jordan
3. Jonathan Michael Borwein
4. Jelena Diakonikolas
5. Constantinos Daskalakis
6. Simeon Reich
7. Zeyuan Allen-Zhu
8. Shoham Sabach

In Geometry and Topology

1. J. Vanderwerff
2. Michael I. Jordan
3. Jonathan Michael Borwein

In Mathematical Physics

1. J. Vanderwerff
2. Michael I. Jordan
3. Jonathan Michael Borwein
4. Constantinos Daskalakis

Citations in the corpus

• Efficient Methods for Structured Nonconvex-Nonconcave Min-Max Optimization (2020, background): “It is well known that no function can be simultaneously smooth and strongly convex w.r.t. an ℓ p norm with p = 2 and have a condition number independent of the dimension [8].”
• Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “The material for the last two subsections of this section is based largely on [86].”
• Curvature of Feasible Sets in Offline and Online Optimization (2020, unknown): “It is known that 2-convex norms ‖·‖ have its square ‖·‖2 being mid-point strongly convex [4, 5, 7, 9, 34].”