[Paper Review] Every non-smooth $2$-dimensional Banach space has the Mazur-Ulam property

Тарас Банах, Javier Cabello Sánchez|arXiv (Cornell University)|Mar 16, 2021

Advanced Banach Space Theory21 references10 citations

TL;DR

This paper proves that every non-smooth 2-dimensional Banach space satisfies the Mazur–Ulam property: any isometry between its unit sphere and the unit sphere of another Banach space extends to a linear isometry of the underlying spaces. The proof uses natural parameterization of the unit sphere, analyzes one-sided derivatives and radial/tangential jumps at non-smooth points, and leverages isometric invariance of intrinsic metrics and derivative jump structures to construct the linear extension.

ABSTRACT

A Banach space $X$ has the $Mazur$-$Ulam$ $property$ if any isometry from the unit sphere of $X$ onto the unit sphere of any other Banach space $Y$ extends to a linear isometry of the Banach spaces $X,Y$. A Banach space $X$ is called $smooth$ if the unit ball has a unique supporting functional at each point of the unit sphere. We prove that each non-smooth 2-dimensional Banach space has the Mazur-Ulam property.

Motivation & Objective

• To resolve Tingley’s Problem in the 2-dimensional case by establishing the Mazur–Ulam property for non-smooth Banach spaces.
• To extend the known result that non-strictly convex 2D Banach spaces have the Mazur–Ulam property to the non-smooth case.
• To characterize the behavior of isometries on unit spheres of non-smooth 2D Banach spaces using geometric and analytic tools.
• To prove that isometries between unit spheres of non-smooth 2D spaces extend linearly by analyzing radial and tangential jumps in the natural parameterization.

Proposed method

• Use the natural parameterization r: R → SX of the unit sphere to represent points by arc length, ensuring ∥r′(s)∥ = 1 almost everywhere.
• Define radial and tangential jumps ˙(s) and ¨(s) at points where r′ has a discontinuity, capturing non-smoothness in the sphere.
• Analyze the asymptotic behavior of distances under isometries using the derivative jump structure, particularly near non-smooth points.
• Show that isometries preserve the intrinsic metric on half-spheres and induce isometries on parameterized intervals [0, L], leading to a unique linear extension.
• Construct a linear operator I: X → Y by setting I(e1) = f(e1), I(e2) = f(e2) where e1 is a non-smooth point and e2 is aligned with the average derivative at e1.
• Prove that the derivative structure (radial and tangential jumps) is preserved under isometry, ensuring I(r′(s)) = ˜r′(s) for all s, which implies linearity of the extension.

Experimental results

Research questions

• RQ1 Does every non-smooth 2-dimensional Banach space have the Mazur–Ulam property?
• RQ2 Can an isometry between the unit spheres of two 2D Banach spaces be extended to a linear isometry if the domain space is non-smooth?
• RQ3 How do radial and tangential jumps in the natural parameterization of the unit sphere relate to isometric extensions?
• RQ4 What role does the intrinsic metric on the upper half-sphere play in characterizing isometric extensions?
• RQ5 Is the linear extension of a sphere isometry uniquely determined by the behavior at non-smooth points and derivative jump invariants?

Key findings

• Every non-smooth 2-dimensional Banach space has the Mazur–Ulam property, completing a duality to Theorem 1.3 on non-strictly convex spaces.
• The natural parameterization r: R → SX satisfies ∥r′(s)∥ = 1 almost everywhere and is non-expanding, enabling arc-length-based analysis.
• At non-smooth points, the radial jump ˙(s) ≤ 0 and |¨(s)| < 1, with r′(s) discontinuous iff ˙(s) ≠ 0.
• Isometries between unit spheres preserve the intrinsic metric on upper half-spheres, and the parameterization ensures that f↾˘SX is isometric to ˜r↾[0,L].
• The linear operator I defined by I(e1) = f(e1), I(e2) = f(e2) extends f to a linear isometry of X and Y, with I(r(s)) = f(r(s)) for all s.
• The derivative jump invariants (˙(0), ¨(0)) are preserved under isometry, ensuring I(r′(s)) = ˜r′(s) for all s ∈ (0, L), which implies linearity of the extension.