Does the unit sphere remember the whole Banach space?

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Family 322 · Functional analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Suppose you know only the unit sphere of a normed vector space, together with distances measured by the ambient norm. You are given a distance-preserving bijection from that sphere to the unit sphere of another space. Does it necessarily come from a linear isometry of the entire spaces?

This is Tingley’s problem. The proposed extension is easy to write: keep the distance from the origin and apply the sphere map to the direction. What is difficult is proving that this radial extension respects distances between points at different radii, and hence respects addition.

Family 322 claims a positive solution for all nonzero real Banach spaces. There is no finite-dimensionality, separability, reflexivity, smoothness or strict-convexity assumption. Applied to complex spaces, the conclusion is real linearity, not necessarily complex linearity.

2. Background and essential definitions

A Banach space is a vector space with a complete norm. Its unit sphere is $S_X=\{x:\|x\|=1\}$, with distance $d(x,y)=\|x-y\|$. This is the ambient chord distance, not distance measured by traveling along the sphere.

An isometry $f:S_X\to S_Y$ preserves every pairwise distance. Surjectivity means every target sphere point is reached; together these properties make $f$ a bijection.

If a linear extension exists, it is forced to be

$$T(0)=0,\qquad T(x)=\|x\|f\left(\frac{x}{\|x\|}\right)\quad(x\ne0).$$

This map automatically preserves norms and is homogeneous for positive real scalars. It also preserves distances between points with the same norm, by scaling the sphere isometry. Neither fact alone proves additivity.

A supporting functional at a unit vector is a linear functional $\varphi$ with $\|\varphi\|=1$ and $\varphi(x)=1$. It describes a supporting hyperplane of the unit ball. When two unit vectors share a support, convex combinations between them remain on the same supporting face. Norms with flat faces make this geometry different from a Euclidean sphere.

A space is strictly convex when its sphere has no nontrivial line segment, and smooth when each nonzero vector has a unique norming support. M1 assumes neither property, so supports and directions can have nonunique behavior.

The Mazur–Ulam property for a space means that every surjective sphere isometry from it to any real Banach target extends linearly. This is stronger than an extension theorem restricted to a specified target class.

3. Development history and previous partial results

Mazur and Ulam proved that a surjective isometry between entire real normed spaces is affine. If it fixes zero, it is linear. Mankiewicz extended this kind of result to convex sets with nonempty interiors. A sphere has empty ambient interior, so these theorems do not directly extend a sphere map.

Tingley’s original work established antipodal preservation in finite dimensions: opposite sphere points map to opposite points. That is necessary for a real-linear extension, but it does not by itself recover distances between unequal radii.

Early positive results treated concrete spaces, including continuous-function spaces and sequence spaces. Subsequent work distinguished class-to-class theorems from the arbitrary-target Mazur–Ulam property. Tan proved that property for real $L^p$ spaces over sigma-finite measure spaces in the stated range $1<p<\infty$, $p\ne2$.

Facial geometry supplied another approach. Sphere isometries preserve maximal convex subsets of spheres, allowing support and face structure to constrain possible maps. In operator algebras, Fernández-Polo–Peralta treated von Neumann algebras, Mori treated their preduals, and Mori–Ozawa proved the arbitrary-target property for unital complex C*-algebras viewed as real spaces and for real von Neumann algebras.

Banakh established the property for every two-dimensional real Banach space. Other recent results covered specific sphere self-isometries and noncommutative $L^p$ or Schatten settings. M1 claims to remove all structural restrictions in the Banach-space problem rather than add another special class.

4. What the new manuscripts claim

M1 states that for any nonzero real Banach spaces $X,Y$ and any surjective sphere isometry $f:S_X\to S_Y$, the radial map in Section 2 is a surjective real-linear isometry. It is the unique linear extension of $f$.

There is no dimension or cardinality restriction. The spaces may have flat sphere faces, nonsmooth norms, or nonreflexive behavior. Completeness is included through the Banach-space hypothesis.

For complex spaces regarded as real ones, the theorem guarantees a real-linear extension. It does not guarantee $T(ix)=iT(x)$. Complex conjugation on $\mathbb C$ already illustrates why a metric sphere map need not preserve multiplication by $i$.

The source organizes the key metric obstruction using

$$D_q(x,y)=\|f(x)-qf(y)\|-\|x-qy\|, \qquad 0\leq q\leq1,$$

for $x,y\in S_X$. This measures distortion between two radius levels. It vanishes at $q=0$ and $q=1$, while the unresolved values lie strictly between them.

M1 proves that the maximal absolute defect is zero. Once every $D_q$ vanishes, scaling handles arbitrary pairs of nonzero vectors, and surjectivity supplies the whole-space isometry. The classical midpoint-reflection principle then yields real linearity.

The theorem is exact. It does not state a quantitative stability result for maps that preserve sphere distances only approximately.

5. Why the advance matters

Importance: 5/5. If correct, this resolves a central metric extension problem and establishes that the unit sphere’s ambient metric determines the real linear geometry of the entire Banach space.

The conclusion is stronger than recognizing a normed space up to some nonlinear equivalence. It recovers the unique linear isometry compatible with the specified sphere map. Thus both the object and the actual correspondence are determined.

The absence of regularity assumptions matters. Euclidean intuition relies on roundness and unique normal directions; general Banach balls can have corners, large flat faces and noncompact spheres. M1’s proposed method is designed to work without replacing those spaces by smooth or finite-dimensional models.

6. The main idea

Make a hypothetical largest error occur

Assume some sphere isometry has positive maximal defect

$$M=\sup_{x,y\in S_X,\,0\leq q\leq1}|D_q(x,y)|>0.$$

The supremum might not be attained because unit spheres can be noncompact. M1 passes to a possibly enlarged pair of Banach spaces built from bounded sequences and an ultrafilter. The sphere isometry acts coordinatewise, and its inverse ensures the induced map remains onto.

The construction preserves the universal defect bound while making one positive defect attain $M$. Replacing the original map by its inverse if necessary fixes the sign. This is a contradiction device; it does not claim that every original sphere is compact or that its supremum was attained there.

Propagate extremality through supports

At a maximizing configuration, supporting functionals turn equality and near-equality in triangle inequalities into geometric restrictions. Common supports propagate the extremal configuration along appropriate directions.

The source constructs a nonempty closed bounded convex set of directions and an opposite-endpoint return map preserving it. The goal is to find a fixed direction, which aligns two relevant chords and makes their length relations directly comparable.

Ordinary compactness is unavailable. Instead, M1 estimates the Kuratowski measure of noncompactness, which measures how small finite coverings by bounded-diameter sets can be made. The return map strictly contracts this measure. Darbo’s fixed-point theorem then supplies the aligned direction. It is contraction of noncompactness, not necessarily an ordinary Lipschitz contraction of distances.

Eliminate the aligned configuration

Once the chords are aligned, the manuscript divides the possible parameters into two regimes. In one, a small radial perturbation would produce a defect larger than the supposed maximum. In the other, support and convexity estimates contradict one of the original sphere distances, which $f$ preserves exactly.

Thus the positive maximal defect cannot occur. This is the proof-specific geometric stage: the ultrapower and fixed-point theorems provide the setting, while the invariant direction set, return-map estimate and chord contradiction do the new work.

Recover distances and then addition

For nonzero vectors of norms $r\geq s$, write them as $rx$ and $sy$ with unit vectors $x,y$. The ratio $q=s/r$ lies in $[0,1]$. Vanishing of $D_q$ gives

$$\|T(rx)-T(sy)\|=r\|f(x)-qf(y)\| =r\|x-qy\|.$$

This is the original distance. The radial extension is therefore a surjective isometry fixing zero, and the whole-space Mazur–Ulam argument supplies additivity and real homogeneity. Uniqueness was already forced by the radial formula.

7. Scope, limitations, and open questions

Surjectivity of the sphere map is essential to the stated theorem and to the use of its inverse. M1 does not claim that every distance-preserving embedding of one sphere into another has an onto linear extension.

The conclusion is real linearity even for complex spaces. It concerns Banach spaces, not arbitrary incomplete normed spaces, quasi-Banach spaces, or unrelated metric spheres. Results for those settings need their own hypotheses.

The proof’s passage to enlarged spaces is used only to rule out positive defect. The resulting extension is defined on the original spaces by the explicit radial formula. No additional choice of values away from the sphere is required.

Finally, exact rigidity does not automatically give a bound for approximately distance-preserving maps. Quantitative versions, extension from only part of a sphere, and weaker surjectivity assumptions remain separate questions beyond the manuscript’s main claim.

8. Sources and further reading

• Mori and Ozawa, Mankiewicz’s theorem and the Mazur–Ulam property for C*-algebras, for a broad earlier arbitrary-target theorem.
• Banakh, Every 2-dimensional Banach space has the Mazur–Ulam property, for the complete two-dimensional case.
• M1 explains its defect-maximization, common-support and Darbo fixed-point steps, with the exact Banach-space and real-linearity conventions.

Source manuscripts

• M1. OpenAI, A positive solution to Tingley’s problem (September 23, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.