Geometric Invariants of Surjective Isometries between Unit Spheres

• Main Authors: Almudena Campos-Jiménez, Francisco Javier García-Pacheco
• Format: Article
• Language: English
• Published: MDPI AG 2021-09-01
• Series: Mathematics
• Subjects: tingley problem; Mazur-Ulam property; surjective isometry; extension isometries; geometric invariants; extreme point
• Online Access: https://www.mdpi.com/2227-7390/9/18/2346

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></semantics></math></inline-formula> be Banach spaces and let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>:</mo><msub><mi>S</mi><mi>X</mi></msub><mo>→</mo><msub><mi>S</mi><mi>Y</mi></msub></mrow></semantics></math></inline-formula> be a surjective isometry. The most relevant geometric invariants under surjective isometries such as <i>T</i> are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if <i>F</i> is a maximal face of the unit ball containing inner points, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>(</mo><mo>−</mo><mi>F</mi><mo>)</mo><mo>=</mo><mo>−</mo><mi>T</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We also show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></mrow></semantics></math></inline-formula> is a non-trivial segment contained in the unit sphere such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mo>(</mo><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo><mo>)</mo></mrow></semantics></math></inline-formula> is convex, then <i>T</i> is affine on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></mrow></semantics></math></inline-formula>. As a consequence, <i>T</i> is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property <i>P</i>, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property <i>P</i>.