A convex-valued selection theorem with a non-separable Banach space

Abstract

In the spirit of Michael’s selection theorem [6, Theorem 3.1”’], we consider a nonempty convex-valued lower semicontinuous correspondence φ:X→2Y{\varphi:X\to 2^{Y}}. We prove that if φ has either closed or finite-dimensional images, then there admits a continuous single-valued selection, where X is a metric space and Y is a Banach space. We provide a geometric and constructive proof of our main result based on the concept of peeling introduced in this paper.

Keywords

• barycentric coordinates
• continuous selections
• lower semicontinuous correspondence
• closed-valued correspondence
• finite-dimensional convex values
• separable banach spaces
• 54c60
• 54c65
• 54d65