Abstract A bounded subset of a (finite or infinite dimensional) normed linear space is said to be complete (or diametrically complete) if it cannot enlarged without increasing its diameter. Any contained in set having the same diameter, which called completion . We survey characterizations, basic properties, facts about structure interior and boundary, asymmetry sets. Different methods obtain c...

The paper investigates linear functionals ϕ : ℝn → ℝ preserving a set ℳ⊆ ℝ, i.e., such that ϕ(v) ∈ ℳ for any vector u with components in ℳ. For various types of subsets real numbers, characterizations the preserve them are obtained. In particular, sets ℤ,ℚ, ℤ+,ℚ+,ℝ+, several infinite integers, bounded and unbounded intervals, finite numbers considered. A characterization also allows one to desc...