1 Introduction

This note deals with three problems posed in the 1930s by two prominent members of the Polish school of mathematics. The first problem is known as the isometric Banach conjecture and deals with the simplest characterization of Hilbert spaces among Banach spaces. It was stated in 1932 by Stephan Banach in his book Théorie des opérations linéaires [4]. The second is Problem 68 in the Scottish Book [17], posed by Stanislaw Ulam in 1936, and the third is the well known floating body problem stated by Stanislaw Ulam in 1935 and which is also known as Problem 19 of the Scottish Book.

I don’t remember who told me about or where I first heard of the “Scottish Book”. I only remember being spellbound upon learning of the existence of a notebook of math problems written in a Polish café or bar by mathematicians. I remember hearing that the notebook was buried in a football field during World War II and mysteriously recovered at the end of the war, without it being known at that time where it was or what kind of problems it contained. I imagined Banach, Ulam, and Steinhaus spending their days in a bar, talking about mathematics, just for the pleasure of wasting their lives thinking about it. I also heard that the authors of the problems offered bottles of cognac, whisky, or a simple coffee as a reward to whoever solved one of their problems. The so-called “Scottish Book” is an informal collection of problems in mathematics, written starting around 1935 in the Scottish Café in the city of Lvov, Poland (now Ukraine). Most of the problems were proposed by a small group of local mathematicians, who included Ulam, Banach, Steinhaus, Mazur, Borsuk, Kac, Auerbach, Kuratowski, and Sierpinsky, among others. These mathematicians met on Saturdays for the weekly talk of the Lvov section of the Polish Mathematical Society. Their meetings took place in a seminar room at the University and therefore close to the Scottish Café. Usually the program consisted of talks that lasted four or five minutes; half-hour talks were unusual, and hour-long talks were very rare. Of course, there was some discussion in the seminar room, but the really fruitful discussions took place in the Scottish Café after the meeting was officially over.

It was Steinhaus who discovered Banach; in fact, he used to say that Banach was his greatest discovery. Steinhaus, as a young man, was a teacher in Krakow, a city 300 kilometers west of Lvov. Let Steinhaus tell us about the day he met Banach: “One morning, while walking through a park, I overheard two young students, who were sitting on a bench, arguing about the Lebesgue integral. The Lebesgue integral was then a very new theory. I was so intrigued that I began to chat with the two young men, one of whom was Banach. I was very impressed and since then I supported and advised Banach to continue his studies”. Banach, on the other hand, was a very eccentric person in his habits and in his personal life. He never wrote an exam, because he hated them intensely, but he wrote so many original articles and proposed so many new ideas that years later he received international recognition for his work. Banach used to spend hours and even whole days in the Scottish Café, especially towards the end of the month, when university faculty were paid. One day he decided that since they talked about so many things, they should write down the ideas, when possible, so that they would not forget them. He brought then a suitable notebook in which he began to write the problems. The notebook was kept at the Scottish Café by a waiter who knew the ritual: when Banach, Ulam or Mazur arrived, all they had to do was say, “the book, please”, and the waiter would bring it immediately along with a few cups of coffee. As the years passed, there were more and more problems proposed by other Polish mathematicians; Borsuk, for example, and many others. Another figure of historical importance in the development of mathematics and physics in the twentieth century, prominent in the Los Alamos Manhattan Project, whose life was filmed in the movie “Adventures of a Mathematician” and who was one of the most frequent visitors to the Scottish Café when he was still very young is Stanislaw Ulam. Both Stefan Banach and Stanislaw Ulam are important figures in contemporary mathematics without whom it would be difficult to explain today’s mathematics.

The “Scottish Book" consists of exactly 193 problems, some solved, others still unsolved. The topics it covers are highly varied, but all of them have as a common denominator the simplicity and clarity with which they are laid out. All of them were written in the original notebook, in the Scottish Café. Most of the problems were posed by Banach, Ulam, Steinhaus or Mazur, but there are also such famous names as Erdős, Frechet, Infeld, Kuratowski, Sierpinski, Eilenberg, Lusternik, von Neumann, Knaster, and Alexandroff, among others. The first problem is dated July 17, 1935, and the last is dated May 31, 1941. Many of the problems remain unsolved and the prizes or rewards on offer range from a bottle of champagne, a bottle of whiskey, a bottle of wine, a glass of brandy, a small beer, a cup of coffee, 100 grams of caviar, dinners in various restaurants, up to a kilogram of bacon or a live goose.

2 The isometric conjecture of Banach

In his book Théorie des opérations linéaires (1932) [4] Stephan Banach asked the following question (see [4, p. 244], or p. 152 of the English translation, remarks on Chap. XII, Property 5).

Let V be a Banach space, real or complex, finite or infinite dimensional, all of whose n-dimensional subspaces, for some fixed integer n, \(2\le n<\dim (V)\), are isometrically isomorphic to each other. Is it true that V is a Hilbert space?

It is important to note that Banach’s question is a codimension one problem, since every Banach space, all of whose subspaces of a fixed dimension \(n\ge 2\) are Hilbert spaces, is itself a Hilbert space, which easily follows from the elementary characterization of a norm coming from an inner product via the “parallelogram law." Therefore, an affirmative answer for n in codimension one immediately implies an affirmative answer for n in all codimensions.

2.1 Current state of the conjecture

First of all, the conjecture is not completely solved. It was proved first for \(n=2\) and real V in 1935 by three of the most frequent visitors to the Scottish Café; Auerbach, Mazur and Ulam [2].

The conjecture was proved for all \(n\ge 2\) and infinite dimensional real Banach spaces V in 1959 by A. Dvoretzky [10]. In the same classical paper, an important structural theorem about Banach spaces was proved by answering a question of Alexander Grothendieck. Dvoretzky proved that every sufficiently high-dimensional Banach vector space will have low-dimensional subspaces that are approximately Hilbert spaces or equivalently, that every high-dimensional symmetric convex body has low-dimensional sections that are approximately ellipsoids. This proves the Banach conjecture for infinite dimensional real Banach spaces V.

In 1967, M. Gromov [13] proved the conjecture for even n and all V, real or complex. Also, in this paper Gromov proved the conjecture for odd n and real V with \(\dim (V)\ge n+2\), and for odd n and complex V with \(\dim (V)\ge 2n\). Gromov’s proof of the conjecture for even n and real or complex V uses ideas of topology of the orthogonal groups; in particular the notion of reduction of the structure groups of a fiber bundle. On the other hand, Gromov’s proof of the conjecture for odd n and real V with \(\dim (V)\ge n+2\), and for odd n and complex V with \(\dim (V)\ge 2n\) used different topological ideas; in particular, he used the notion of Whitney-Stiefel manifolds.

In the 1970s, Vitali Milman [18] gave a different proof of Dvoretzky’s theorem that includes the complex case; in particular, proving Banach’s conjecture for infinite dimensional complex Banach space V. Milman’s proof was one of the starting points for the development of asymptotic geometry analysis. Regarding this proof, Gowers [12] claims the following: the full significance of measure concentration was first realized by Vitali Milman in his revolutionary proof of Dvoretzky’s theorem. It is a milestone in the finite-dimensional theory of Banach spaces. While I feel sorry for a mathematician who cannot see its intrinsic appeal, this appeal on its own does not explain the enormous influence that the proof has had, well beyond Banach space theory.

In 2019, fifty years after the work of Gromov and Milman, Luis Hernández Lamoneda and Gil Bor, mathematicians from CIMAT, Guanajuato, and Luis Montejano and Valentín Jiménez, from the Institute of Mathematics of the National University of Mexico (UNAM), decided to introduce the study of the topology of compact Lie groups to the solution of the conjecture by proving it when V is real and n is of the form \(n=4k+1\ge 5\), \(n\ne 133\) [5]. The reason for the strange \(n\ne 133\) exception is that our techniques were not able to be applied to the exceptional Lie group \(E^7\), whose dimension is 133.

A year later, Javier Bracho and Luis Montejano, both from the Institute of Mathematics, proved the conjecture when V is complex and n is of the form \(n=4k+1\ge 5\) [6]. Although this time the compact Lie group topology techniques were better suited to the complex case, the study of sections of a complex body of revolution required new and original ideas.

All these topological techniques were unable to prove this conjecture when V is real or complex and \(n=4k+3\). This includes the cases \(n=3\) and \(n=7\). Because the space of vectors tangent to these two spheres, \(\mathbb S^3\) and \(\mathbb S^7\), is parallelizable, as is well known, algebraic topology is no longer a good tool in these cases. Indeed, recently S. Ivanov, D. Mamaev and A. Nordskova [14] proved the Banach conjecture for \(n=3\) using non-topological ideas. Therefore, the conjecture remains unproven when V is real or complex and \(n=4k+3\), \(k>0\).

2.2 Sketch of the proof when V is real and \(n\equiv 1\) mod 4

The first step is to translate the conjecture into a geometric problem.

First of all, it is well known that a Banach space is a Hilbert space if and only if its unit ball is an ellipsoid, the affine image of a euclidean ball. On the other hand, two Banach spaces \(V_1\) and \(V_2\) are isometric if and only there is a linear map \(\Phi :V_1\rightarrow V_2\) preserving the norm if and only if the unit balls of \(V_1\) and \(V_2\) are linearly equivalent. So, the isometric Banach conjecture over the real numbers is equivalent to the following characterization of the ellipsoid.

Let \(B\subset \mathbb {R}^{N}\) be a symmetric convex body, all of whose sections by n-dimensional linear subspaces, for some fixed integer n, \(1<n<N\), are linearly equivalent. Is it true that B is an ellipsoid?

The fact that a symmetric body is an ellipsoid if and only if all its hyperplane sections through the center are ellipsoid makes the above problem a codimension one problem. So this section is devoted to giving a sketch of the proof of the following theorem:

Theorem 2.1

(Bor–Hernandez–Jimenez–Montejano) Let \(B\subset \mathbb {R}^{n+1}\), \(n\equiv 1, \mod 4\), \(n>5 \ne 133\), be a convex symmetric body, all of whose sections by n-dimensional subspaces are linearly equivalent. Then B is an ellipsoid.

A symmetric convex body \(K\subset \mathbb {R}^n\) is a symmetric body of revolution if it admits an axis of revolution; i.e., a 1-dimensional linear subspace L such that each section of K by an affine hyperplane A orthogonal to L is an \((n-1)\)-dimensional closed euclidean ball in A, centered at \(A\cap L\) (possibly empty or just a point). If L is an axis of revolution of K, then \(L^\perp \) is the associated hyperplane of revolution. An affine symmetric body of revolution is a convex body linearly equivalent to a symmetric body of revolution. The images, under the linear equivalence, of an axis of revolution and its associated hyperplane of revolution of the body of revolution are an axis of revolution and associated hyperplane of revolution of the affine body of revolution (not necessarily perpendicular any more). Clearly, an ellipsoid centered at the origin is an affine symmetric body of revolution and any hyperplane serves as a hyperplane of revolution.

The proof of Theorem 2.1 is in two steps. In the first step we prove that under the hypothesis of the theorem, every hyperplane section through the origin is an affine body of revolution. The second step is devoted to proving that if every hyperplane section through the origin is an affine body of revolution, then at least one of the hyperplane sections through the origin is an ellipsoid. Finally, if one of the hyperplane sections through the origin is an ellipsoid, by hypothesis, all hyperplane sections through the origin are ellipsoids and it is well known that the latter implies that our convex body is an ellipsoid.

It is in the first step where we use the topology of compact Lie groups as follows. Let \(B\subset \mathbb {R}^{n+1}\) be a symmetric convex body, all of whose hyperplane sections are linearly equivalent to some fixed symmetric convex body \(K\subset \mathbb {R}^n\). Let \(G_K:=\{g\in GL_n(\mathbb {R})\vert g(K)=K\}\) be the group of linear symmetries of K. Then it is possible to prove that the structure group of \(S^n\) can be reduced to \(G_K\). For the notion of reduction of the structure group, see [27]. Our next task is to understand the possible reductions of the structure group of \(S^n\), a classical problem in topology. The results we need are contained in the purely topological Theorem 2.2, which implies that K is an affine symmetric body of revolution.

But first, another definition. We say that a subgroup \(G\subset GL_n(\mathbb {R})\) is reducible if the induced action on \(\mathbb {R}^n\) leaves a k-dimensional linear subspace invariant, \(1<k<n\); otherwise, it is an irreducible subgroup of \(GL_n(\mathbb {R})\). (Beware of the potentially confusing use of the notions ‘reducible’ and ‘can be reduced’ in the statement of the following theorem.)

Theorem 2.2

(Bor–Hernandez–Jimenez–Montejano) Let \(n\equiv 1\mod 4, n\ge 5\), and suppose that the structure group of \(S^n\) can be reduced to a closed connected subgroup \(G\subset SO_n\). Then:

  1. (a)

    If G is reducible, then it is conjugate to a subgroup of the standard inclusion \(SO_{n-1}\subset SO_n\), acting transitively on \(S^{n-2}\).

  2. (b)

    If G is irreducible, then \(G=SO_n\), or \(n=133\) and \(G\subset H\subset SO_{133}\), where H is the adjoint representation of the simple exceptional Lie group \(E_7\).

Remember that the second step is devoted to proving that if every hyperplane section through the origin is an affine body of revolution, then at least one of the hyperplane sections through the origin is an ellipsoid. This part of the proof is perhaps the most original. It is based on a successful combination of ideas from convex geometry and algebraic topology; in particular, from a careful study of bodies of revolution. \(\square \)

3 Problem 68 of the Scottish Book

In [20], Montejano gave a complete solution to Problem 68 of the Scottish Book posed by Ulam in 1936, which reads as follows.

“There is given an n-dimensional manifold R with the property that every section of its boundary by a hyperplane of \(n-1\) dimensions gives an \((n-2)\)- dimensional closed surface (a set homeomorphic to a surface of the sphere of this dimension). Prove that R is a convex set.”

We may interpret this problem as follows: “If R is a compact, \((n+1)\) -dimensional manifold with boundary in \(\mathbb {R}^{n+1}\) for which every n -dimensional hyperplane H that meets R in more than one point has \(H\cap \partial R\) homeomorphic to an \((n-1)\)-sphere, is R convex?"

In 1983, Schreier [26] showed first that a two-dimensional surface in \(\mathbb {R} ^{3}\), each of whose nondegenerate planar sections is a Jordan curve, is the boundary of a convex body. Then later, in 1990, Montejano [20] generalized Schreier’s theorem as follows.

Theorem 3.1

Let N be a closed, connected n-manifold topologically embedded in \(\mathbb {R}^{n+1}\). Suppose that for every n -dimensional hyperplane H that meets N in more than one point, \(H-N\) has exactly two components. Then N is the boundary of a convex \((n+1)\)-body."

As a corollary of this theorem, Problem 68 was settled affirmatively.

Furthermore in the same paper, the following interpretation of Ulam’s problem was also proved in [20]: Let \(1\le k\le n.\) If R is a compact \((n+1)\)-manifold with boundary in \(\mathbb {R}^{n+1}\) for which every k-dimensional plane H that meets the interior of R has \(H\cap \partial R\) a \((k-1)\)-sphere, then R is a convex \((n+1)\)-body.

In [20], the following homological characterization of convexity, which is closely related to Problem 68, is also considered: Let \(1\le k\le n\), and let K be a compact subset of \(\mathbb {R}^{n+1}\). Suppose that for every k-dimensional plane H, \(H\cap K\) is acyclic. Then K is convex. This theorem was also obtained by Kosinski [15] and generalizes the following result of Aumann [3], mentioned in the first edition of [17]: A compact connected set K in \(\mathbb {R}^{3}\) is convex provided that for each plane P, \(P\cap K\) and \(P-K\) are connected; a continuum K in \(\mathbb {R}^{n}\) is convex if and only if \(\mathbb {R}^{n}-K\) is connected and the intersection of K with each \((n-1)\)-dimensional hyperplane is convex; and a closed subset K of \(\mathbb {R}^{n}\) is convex if and only if \(K\cap P\) is simply connected for all 2-dimensional hyperplanes P.

Later Montejano [22] and Shchepin [23] gave homological characterizations of convex sets in terms of acyclic support sets. The ideas of this last series of papers are part of a new branch of tomography called topological tomography. Evgeny Shchepin worked at the Cuernavaca Institute of Mathematics, UNAM, during the last years of the twentieth century. Indeed, he was an important figure in the beginnings of this institution. For a biography of E.V. Shchepin, see [9].

4 Problem 19 of the Scottish Book

S. M. Ulam in his book A collection of Mathematical Problems [29] stated the following problem.

If a body rests in equilibrium in every position on a flat horizontal surface, must it be a sphere?

If by “body” we understand a compact connected subset of \(\mathbb {R}^{n+1}\), L. Montejano [19] proved in 1974 that the shell of a body that rests in equilibrium in every position on a flat horizontal surface is an n-sphere.

Indeed, this is the density zero case of the floating body problem, also known as Problem 19 of the Scottish Book, stated by Ulam in 1955.

Is a solid of uniform density which floats in water in equilibrium in every position a sphere?

If the body is centrally symmetric and its density is 1/2 (that is, the area submerged under water is always half the area of the body), then there are no solutions to Problem 19 other than the sphere. The proof is a nice exercise of Fourier analysis in spherical integration. See Falconer [11], Schneider [25] or Section 13.3.1 of [16].

In 2022, more than 85 years after Problem 19 was posed, Ryabogin [24] proved that there exists a 3-dimensional convex body of density 1/2 which is not a sphere, yet floats in equilibrium in every position. It is interesting to note that although this body cannot be symmetric, it is a body of revolution.

The two-dimensional version of the problem concerns a cylinder of uniform density \(\rho \) that floats in water in equilibrium in every position, having its axis parallel to the water surface. This problem is connected to several dynamical systems, such as for example the tire track problem [28], the problem of the existence of closed carousels [7], and the problem of determining the trajectory of a charge moving in a perpendicular parabolic magnetic field.

As in dimension 3 in the case of density \(\rho =\frac{1}{2}\), if the figure is centrally symmetric, then there are no solutions other than the circular cylinder. Nevertheless, a regular customer at the Scottish Café, H. Auerbach [1], showed that in the case of density \(\rho =\frac{1}{2}\), a cylinder of uniform density \(\rho \) which floats in water in equilibrium in every position, having its axis parallel to the water surface, need not be circular or even convex, and gave a class of examples. All these examples coincide with the Zindler curves [31] which, with a suitable geometric construction, are in one-to-one correspondence with the family of the figures of constant width. See Section 5.4 of [16].

If a figure F of density \(\rho \) floats in equilibrium in every position, then the water surface divides the boundary of F in constant ratio, say, \(\sigma :1-\sigma \). We call \(\sigma \) the perimetral density of D. In 2004, Javier Bracho, Luis Montejano and Deborah Oliveros, all three from the UNAM Institute of Mathematics [8], proved that if the perimetral density \(\sigma \) is \(\frac{1}{3}\), \(\frac{1}{4}\), \(\frac{1}{5}\), or \(\frac{2}{5}\), then the solution is a circular cylinder. Indeed, apart from the symmetric case of density \(\frac{1}{2}\), these are among the few known results where the conjecture is true. The proof is based on the study of an interesting type of dynamical system called carrousels (see [7] and [21]), defined by the authors to understand the floating body problem in dimension 2. In contrast, Wenger [30] was able to obtain noncircular solutions for \(\rho \not =\frac{1}{2}\not =\sigma \).