Abstract
We introduce the circle-uniqueness of Pythagorean orthogonality in normed linear spaces and show that Pythagorean orthogonality is circle-unique if and only if the underlying space is strictly convex. Further related results providing more detailed relations between circle-uniqueness of Pythagorean orthogonality and the shape of the unit sphere are also presented.
Highlights
We denote by X = (X, ‖ ⋅ ‖) a real normed linear space whose dimension is at least 2
We introduce the circle-uniqueness of Pythagorean orthogonality in normed linear spaces and show that Pythagorean orthogonality is circle-unique if and only if the underlying space is strictly convex
The following lemma concerning the intersection of two circles in a Minkowski plane is one of our main tools
Summary
James proved the line-existence of Pythagorean orthogonality: for each pair of vectors x and y in X, there exists a number α such that x⊥Pαx + y. James proved that in each line parallel to the line ⟨−x, x⟩ there exists a vector that is Pythagorean orthogonal to x.
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