Tingley’s sphere-isometry problem

The explanation

Draw a square with corners at (1, 1), (1, −1), (−1, −1) and (−1, 1). Now measure length by an unusual rule. The length of (a, b) is the larger of |a| and |b|. Under this rule, every point on the square's edge has length exactly 1. So the square plays the role of a circle. Mathematicians call it the unit sphere, meaning the set of all points of length 1. Distances follow the same rule. The corner (1, 1) and the point (1, 0) are 1 apart. The points (1, 0) and (1, 0.5) are 0.5 apart.

Now take a map that moves the points of the edge around. Suppose it keeps every distance between edge points, and it reaches every point of the edge. Such a map is called a surjective isometry of the sphere. A quarter turn of the square is one example. It sends (1, 0) to (0, 1). Tingley's problem asks a simple question. Does every such map come from a linear map of the whole space? Linear means the map respects adding points and stretching them by a number.

There is only one possible candidateSource 1. Take any point that is not zero, such as (0, 0.5). Its length is 0.5, and its direction on the sphere is (0, 1). Move the direction with the given map and keep the length. The paper calls this the radial map. It keeps lengths. It also keeps distances between two points of the same lengthSource 2. The hard part is two points of different lengths. The point (1, 0) has length 1, and (0, 0.5) has length 0.5. Under the square rule they are 1 apart. An ordinary ruler would give about 1.118. The sphere by itself never shows that distance directly. The paper names this as the difficulty: such distances must be recovered from the sphere aloneSource 3.

Older results stop short of the sphere. In 1932, Mazur and Ulam proved a result about distance-keeping maps of a whole real space onto another. Every such map is affine, meaning linear plus a shift. In 1972, Mankiewicz extended this to convex sets with a nonempty interior. A sphere has no interior, so his theorem does not directly give the answerSource 4. Tingley asked the sphere question in 1987. He showed, in finite dimensions, that such maps send opposite points to opposite points.

Later work settled the question space by space. Ding treated sequence spaces of ℓ^p type in 2003, with p above 1 and not equal to 2. Tan treated real L^p spaces in 2012, again with p above 1 and not equal to 2. Others handled operator algebras and other concrete families. In 2022, Banakh settled every two-dimensional space. So the square above was already covered. These results covered particular spaces or families. This paper claims the general case.

This paper proves the full answer, and a Lean proof is listed. It covers every nonzero real Banach space. That is a space of vectors with a length rule and no missing limit points. Every surjective isometry between two such unit spheres extends to the whole spaces. The extension is the radial mapSource 5. It is real-linear, meaning linear when stretching by ordinary real numbers. It reaches every point, keeps all distances, and is the only linear extension. The dimension can be anything, even infiniteSource 6. The sphere may have corners and flat sides, like the square.

The proof works by contradiction. It measures the largest error the radial map could make between two radius levelsSource 7. Suppose that error were above zero. The proof first makes the worst error actually occur, by passing to larger spaces if neededSource 8. It then lines up two extreme chords, using a fixed-point theorem. Convexity estimates then show that this worst case cannot exist. A final step shows the radial map is linear.

So, for the square, the theorem guarantees the distance 1 between the images of (1, 0) and (0, 0.5). More broadly, distances on the unit sphere alone fix the whole space, up to a linear map that keeps lengths. Two limits matter. The map must reach every point of the target sphere. And some spaces let vectors be stretched by complex numbers. For those, treated as real spaces, the result gives real linearity, not complex linearitySource 9.

How sure is this? The Lean scope note lists a formal version of the main theorem. It states the unique surjective real-linear extension for arbitrary nonzero real Banach spaces. It also lists a Comparator statement for that theorem. No specialist review has been published yet.

What it doesn't show. This does not give complex linearity for complex spaces; it gives real linearity only.