For r > 0 a non-empty subset U of a linear space is said to be absolutely r-convex if x, y ∈ U and |λ|r + |μ|r ≤ 1 together imply λx + μy∈ U, or, equivalently, xl, …, xn∈ U and
It is clear that if U is absolutely r-convex, then it is absolutely s-convex whenever s < r. A topological linear space is said to be r-convex if every neighbourhood of the origin θ contains an absolutely r-convex neighbourhood of the origin. For the case 0 < r ≤ 1, these concepts were introduced and discussed by Landsberg(2).