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Continuous interior selections in nonnormable spaces
We prove a selection theorem for convex-valued lower semicontinuous mappings Fto Fréchet spaces under the assumption that every value of Fcontains all interior (in the convex sense) points of its own closure. This is an extension of E. Michael’s theorem 3.1''' in [7]for mappings to Banach spaces. The desired continuous single-valued selection is constructed as a pointwise barycenter mapping with respect to a suitable family of probability measures concentrated on values of F. As an application, we show that, for any metric space Mthere is a continuous mapping which to every compact set K⊂Massigns a probability measure whose support coincides with K. Earlier this fact was proved for a complete separable metric space Mby using a fundamentally different technique based on Milyutin mappings.