## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Workshop/Miniconference on Functional Analysis and Optimization. Simon Fitzpatrick and John Giles, eds. Proceedings of the Centre for Mathematical Analysis, v. 20. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1988), 34 - 38

### Differentiability of distance functions in normed linear spaces with uniformly Gateaux differentiable norm

#### Abstract

Consider a non-empty proper closed subset K of a Banach space $(X, \| \cdot \|)$ where the norm is uniformly Gateaux (uniformly Frechet) differentiable. Then the associated distance function d is guaranteed to be Gateaux differentiable on a dense subset D of X\K. Furthermore, Gateaux (Frechet) differentiability of the distance function at a point $x \in$ X\K can be characterised in terms of the weak* sequences ${d' (x_n)}$ where ${x_n}$ is a sequence in D converging to x.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416340100

**Mathematical Reviews number (MathSciNet)**

MR1009591

**Zentralblatt MATH identifier**

0673.41036

#### Citation

Davis, Brett. Differentiability of distance functions in normed linear spaces with uniformly Gateaux differentiable norm. Workshop/Miniconference on Functional Analysis and Optimization, 34--38, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1988. https://projecteuclid.org/euclid.pcma/1416340100