## Remotal sets in vector valued function spaces.(English)Zbl 1118.46026

Let $$X$$ be a Banach space. A bounded set $$E \subset X$$ is said to be remotal if for every $$x \in X$$ there exists a $$z \in E$$ such that $$\| x-z\| = \sup \{\| x-y\| : y\in E\}$$. This paper deals with remotal sets in the space of Bochner integrable functions $$L^1([0,1],\lambda,X)$$ of the form $$L^1([0,1],\lambda,E)$$ for a remotal set $$E \subset X$$, where $$\lambda$$ denotes the Lebesgue measure. Clearly, since remotal sets are bounded, one cannot expect such a result when the measure has infinitely many atoms (Theorem 2.2), unless $$E = \{0\}$$. The authors first obtain a distance estimate, $$\sup \{\| f-g\| : g \in L^1([0,1],\lambda,E)\} =\int \sup\{\| f(s)-e\| : e \in E\}\,d\lambda(s)$$ and use it to show that when $$E$$ is a closed and bounded convex set such that its closed linear span is finite-dimensional then $$L^1([0,1],\lambda,E)$$ is a remotal set.

### MSC:

 46B20 Geometry and structure of normed linear spaces 46E40 Spaces of vector- and operator-valued functions

### Keywords:

remotal set; space of Bochner integrable functions