### Approximate selections in ${\alpha}$-convex metric spaces and topological degree

#### Abstract

The existence of continuous approximate selections is proved for a class

of upper semicontinuous multifunctions taking closed $\alpha$-convex

values in a metric space equipped with an appropriate notion

of $\alpha$-convexity.

The approach is based on the definition of pseudo-barycenter of an ordered

$n$-tuple of points.

As an application, a notion of topological degree for a class of $\alpha$-convex

multifunctions is developed.

of upper semicontinuous multifunctions taking closed $\alpha$-convex

values in a metric space equipped with an appropriate notion

of $\alpha$-convexity.

The approach is based on the definition of pseudo-barycenter of an ordered

$n$-tuple of points.

As an application, a notion of topological degree for a class of $\alpha$-convex

multifunctions is developed.

#### Keywords

Continuous approximate selection; multifunction; convex metric space; pseudo-barycenter; topological degree

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