### A Cone of Inhomogeneous Second-Order Polynomials.

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Let be a strictly convex separable Banach space of dimension at least 2. Let K() be the space of all nonempty compact convex subsets of endowed with the Hausdorff distance. Denote by ${K}^{0}$ the set of all X ∈ K() such that the farthest distance mapping $a\mapsto {M}_{X}\left(a\right)$ is multivalued on a dense subset of . It is proved that ${K}^{0}$ is a residual dense subset of K().

In the present paper we answer two questions raised by Barbilian in 1960. First, we study how far can the hypothesis of Barbilian's metrization procedure can be relaxed. Then, we prove that Barbilian's metrization procedure in the plane generates either Riemannian metrics or Lagrance generalized metrics not reducible to Finslerian or Langrangian metrics.

It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.