Some research problems

Abstract

Let d(x,y) be a metric in the n-dimensional vector space Rⁿ (without any connection to the metric induced by the norm and the linear operations in Rⁿ). We say that the metric d is invariant with respect to translations if d(x + a, y + a) = d(x,y) for any a, x, y ∈ Rⁿ. Furthermore, we say that a metric d is normable if there exists a norm ∥ · ∥ in Rⁿ such that d(x,y) =∥ x − y ∥ for any x, y ∈ Rⁿ. Finally, we say that a metric d is bounded if the set B = { x ∈ Rⁿ : d(o, x) ≤ 1 { is bounded in Rⁿ . The problem is to describe a condition under which a metric d in Rⁿ is normable.