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Home , Hugo Steinhaus, Juliusz Schauder, Kazimierz Kuratowski, Scottish Book, Stanislaw Ulam

The . from the Scottish Caf´e with selected problems from the new Scottish Book. Second edition. Edited by R. Daniel Mauldin Birkh¨auser/Springer,Cham, 2015 xvii+322 pp. ISBN: 978-3-319-22896-9; 978-3-319-22897-6

REVIEWED BY SERGEI TABACHNIKOV

The genre of this book is rather uncommon: it is a collection of research problems, nearly 200 altogether. Some of the problems are supplied with comments, from very brief to detailed ones spanning several pages. Among recent publications, a remotely similar book is “Arnold’s Problems” [1] (re- viewed by myself in the Winter 2007 issue of this magazine) which is a much larger collection of problems and which spans a much longer period of time (about 50 years). The Scottish Book is also a memorial to the members of the vibrant mathematical community, that existed in then Polish city of Lvov (a.k.a. , Lw´ow,Lemberg; I use the spelling of [6]) between the two World Wars and that perished as a result of WW2. The reader will recognize the names of some of the members of the Lvov school of mathematics: Stefan Ba- nach, , , Stanislaw Mazur, Wladyslaw Or- licz, , , ... For the remarkable and tragic history of this mathematical school see [6] (reviewed in the Winter 2017 issue of this magazine); I shall only briefly describe relevant facts. Located close to the university, a popular gathering place was the Scot- tish Caf´e(see Figure 1); it played the role of a faculty lounge and a faculty club for Lvov . The lively mathematical conversations, that sometimes lasted for hours, included discussion of open problems. For solu- tions of some problems, prizes were offered of impressive diversity: from a bottle of wine or a glass of beer to a bottle of whiskey of measure > 0 (offered

1 Figure 1: Left: the building of the Scottish Caf´e. A copy of the Scottish Book is held in the renovated caf´e. Right: Stanislaw Mazur delivers a live goose to Per Enflo. by ) to 100 grams of caviar, or one kilo of bacon, or a live goose (this problem of Mazur was solved by Enflo in 1972, and the prize was delivered, see Figure 1).

Figure 2: , Hugo Steinhaus, and Stanislaw Ulam.

In summer of 1935, Banach’s wife bought a notebook that was kept at the caf´efor the mathematicians to record the problems. Thus the Scottish Book came into being. The first entry, by Banach, was made on July 17, 1935, and the last one, by Steinhaus, on May 31, 1941. The majority of entries were made in 1935, and many problems were discussed before that. Part of the problems were

2 offered by visitors, in particular, after Lvov came under the Soviet control in September of 1939, by Russian mathematicians: Alexandroff, Bogolubow, Lusternik, Sobolew (again using the spelling of [6]). Concerning the number of the proposed problems, the record holder is S. Ulam, followed by S. Mazur, S. Banach, and H. Steinhaus, see Figure 2 (some problems have multiple authors). The book under review conveniently has the index of the authors of the problems and the problem subject index. Miraculously, the Scottish Book had survived the war. Steinhaus made a typewritten copy of the book and, in 1956, sent it to Ulam, who was working at the Los Alamos National Laboratory at the time. Ulam translated the book into English, made about 300 copies of it, and distributed it among colleagues. In 1979, a conference devoted to the Scottish Book was held at North Texas State University, and the first edition of the book under review followed in 1981. Sadly, a number of authors of the book did not survive the war. On June 30, 1941, Lvov was occupied by the German army, and in July, a number of Polish academics were killed by the Nazi forces (in an action against Polish intelligentsia and leaders of the Polish society). This massacre and the elimination of the Jews1 claimed the following victims among the authors of the Scottish Book: H. Auerbach, M. Eidelheit, S. Ruziewicz, J. Schauder, J. Schreier, L. Sternbach. S. Banach survived the war, but died of cancer in August of 1945. M. Kac and S. Ulam were lucky to have moved to the USA; H. Steinhaus, who was Jewish, managed to survive by changing the name and going into hiding. Before discussing the content of the book in any detail, let me mention that some problems from the Scottish Book or variations on their themes can be found in Ulam’s book [19] and in the expository books by Steinhaus [15, 16]. The book under review comprises four parts. The first consists of five articles based on the talks given at the 1979 conference, by Ulam (“An anec- dotal history of the Scottish book”), Kac (“A personal history of the Scot- tish book”), Zygmund (“Steinhaus and development of Polish mathemat- ics”), Erd´os(“My Scottish Book ‘problems’ ”), and Granas (“KKM-maps”; KKM stands for Knaster–Kuratowski–Mazurkiewicz). The second part is the Scottish Book per se, supplied with the subject index and the index of

1Pre-war population of Lvov was about 30% Jewish; almost all Jewish inhabitants were killed.

3 the authors. The third and fourth parts concern the New Scottish Book. After the war, the borders of moved to the West. In particular, Lvov was incorpo- rated into the , and the Silesian capital of Breslau/Wroclaw became part of Poland. Most of the faculty of the Lvov University moved to Wroclaw, and the New Scottish Book was started in 1946 as a continuation of the original one (the first entry was made by Steinhaus). The New Scottish Book contains nearly 1000 problems, and the book under review presents a selection of these problems, along with the article “Lw´owof the West” by Biler, Krupski, Plebanek, and Woyczy´nski. Naturally, many problems in the Scottish Book concern the subjects that were actively developed by the participants of the Lvov School: Banach spaces, convex , point , geometric topology, , measure theory, , and theory of functions. Not being an expert, it is my impression that commentaries to more problems could be made by a variety of mathematicians, and I hope that in the next edition of the book more problems will be annotated. For example, consider problem 76 by Mazur:

Given in 3-dimensional is a convex surface W and point O in its . Consider the set V of all points P defined by the property that the length of the interval PO is equal to the area of the plane section of W through O and perpendicular to this interval. Is the set V convex?

This problem concerns what is known as intersection bodies. The answer is negative in general, but if W is centrally symmetric with respect to O, then V is convex. This follows from a theorem of Busemann, see section 8.1 of [8] for this material. Another example, problem 147 by Auerbach and Mazur:

Suppose that a billiard ball issues at the angle 45◦ from a corner of the rectangular table with a rational ratio of the sides. After a finite number of reflections from the cushion will it come to one of the remaining three corners?

This is easily proved by developing the trajectory into a straight line (a standard technique in the study of polygonal billiards; cf. item 26 of [15]); I leave the proof as a challenge to the reader.

4 Some problems led to whole areas of research; I guess that the record holder is problem 123 by Steinhaus:

Given are three sets A1,A2,A3 located in the 3-dimensional Eu- clidean space and with finite Lebesgue measure. Does there exist a plane cutting each of the three sets A1,A2,A3 into two parts of equal measure? The same for n sets in n-dimensional space. This is, of course, the famous whose variations and generalizations is a subject of a great contemporary interest. Let me mention a few other problems; the choice, obviously, reflects my mathematical taste. Problem 59 by Ruziewicz: Can one decompose a square into a finite number of squares all different? This well known problem has an unexpected relation to electrical net- works; it led to much research, and it is amply commented upon by P. J. Federico. Problem 18 by Ulam: Let a steady current flow through a curve in space which is closed and knotted. Does there exist a line of force which is also knotted? This problem is solved and commented upon by G. Minton. The answer is in the affirmative, and the proof is computer assisted [14]. Problem 38 by Ulam: Let there be given N elements (persons). To each element we attach k others among the given N at random (these are friends of a given person). What is the probability PkN that from every element one can get to every other element through a chain of mutual friends? (The relation of friendship is not necessarily symmetric!) Find limN→∞ PkN (0 or 1?). The solution to this problem, in the case when the friendship relation is symmetric, is provided in the comment by Erd˝os:the answer is 1 if k 2 and 0 if k = 1. As R. Graham says in his comment, this problem “foreshadowed≥ the emergence of the subject of the evolution of random graphs, which was pioneered by Erd˝osand Renyi more than two decades later.” In case you are interested which problem merited live goose, it is problem 153:

5 Given is a continuous function f(x, y) defined for 0 x, y 1 and ≤ ≤ the number ε > 0; do there exist numbers a1, . . . , an; b1, . . . , bn; c1, . . . , cn with the property that n

f(x, y) ckf(ak, y)f(x, bk) ε | − | ≤ k=1 X in the interval 0 x, y 1? ≤ ≤ As I mentioned, it was solved by Enflo; the answer is negative. Finally, let me mention a problem in whose study I was was involved. It is problem 19 by Ulam: Is a solid of uniform density which will float in water in every position a sphere? This question can be asked in every dimension; substantial, albeit partial, results are available only in dimension 2 (one may think of a floating log), and what follows concerns this case. The answer depends on the relative density of the body. For example, the case of density 1/2 is very flexible: the solutions parameterized by functions of one variable [2] (similarly to figures of constant width). The cases of some other densities are rigid: only a circle is the solution. Unexpectedly, this flotation problem is closely related with another prob- lem, that of bicycle kinematics. One models bicycle as an oriented segment of fixed length that can move in the plane so that the velocity of its rear end is always aligned with the segment: the rear wheel is fixed on the frame, whereas the front wheel can steer. There are many questions one can ask about this model; see a survey in [7]. One of the questions is as follows: suppose you see two tracks, those of front and rear wheels. Can you tell which way the bicycle went? One can call this the Sherlock Holmes problem: the famous detective had to consider it in “The Adventure of the Priory School” mystery, and his argument was erroneous: it correctly determined which tracks were front and rear, but not the direction of the motion; see [10]. In fact, given two closed tire tracks, the front and the rear, usually one can determine the direction of the motion, but sometimes one cannot do it. A trivial example is two concentric circles; are there other examples? It turns out that this problem is equivalent to the 2-dimensional case of Ulam’s flotation problem: the front track is the boundary of the floating

6 Two dimensional bodies which can float in all directions are given by ψper = 2π/n, thus for m = 1 and sufficiently small #. In this limit the δu can be determined from eq. (141) with

n2 n nδz ζ(ω! + δz)= (ω! + δz) ni tan( )+O(q), (199) 12 − − 2 2 2 2 2i inδz nδz n (ω!+δz) /24 σ(ω! + δz)= e− cos( )e + O(ˆq), (200) nqˆ 2

2 where eqs. (349) and (355) and π = n, η3 = n have been used. Then eq. ω3 ω3 12 (141) yields tan(n δu )=n tan( δu ), (201) in agreement with the results obtained in refs. [7, 12, 8], where δu corresponds to π δ and in ref. [6], where δu corresponds to πρ. 2 − 0 Two dimensional bodies which can float in all directionsA are few given cross-sections by ψper = of the bodies are shown in figs. 10 to 23. For odd n the 2π/n, thus for m = 1 and sufficiently small #. In thisinnermost limit the envelopeδu can be corresponds to density ρ =1/2. determined from eq. (141) with n2 n nδz ζ(ω! + δz)= (ω! + δz) ni tan( )+O(q), (199) 12 − − 2 2 2 2 2i inδz nδz n (ω!+δz) /24 σ(ω! + δz)= e− cos( )e body, and+ O the(ˆq), rear track(200) is the envelope of the waterline as the body rotates. nqˆ 2 See Figure 3. 2 where eqs. (349) and (355) and π = n, η3 = n haveAs been the used. body rotates,Then eq. the length of the arc of its boundary, bounded by the ω3 ω3 12 (141) yields waterline remains the same, and the ratio of this length to the total length tan(n δu )=n tan( δu ), of the boundary is a(201) parameter that plays the role of the relative density of the body. If the density is 1/2, this ratio also equals 1/2 but, in general, it in agreement with the results obtained in refs. [7, 12, 8], where δu corresponds isFig. not equal10 m/n to the=1 density./3, OneFig. can prove 11 m/n that if=1 this/3 ratio, of lengthsFig. equals 12 m/n =1/3, to π δ and in ref. [6], where δu corresponds to πρ. 2 0 # =0.1 # =0.2 # =0.5 A− few cross-sections of the bodies are shown in figs.1/3 10 and to 23. 1/4, For then odd an circlethe is the unique solution [4, 5, 17]. innermost envelope corresponds to density ρ =1/2.

Fig. 10 m/n =1/3, Fig. 11 m/n =1/3, FigureFig. 3: 12 Somem/n of=1 Wegner’s/3, solutions: the outer black curve is the front bi- Fig. 13 m/n =1/4, Fig.∗ 14 Fig.∗ Fig.17 15 m/n =1/4, Fig.∗ 18 # =0.1 # =0.2 cycle# track,=0.5 the red inner curveFig. is 16 them/n rear=1 track,/5, the blue and green segments m/n =1/4, # =0.1 m/n =1/5, ! =0.1 m/n =1/5, ! =0.1 are# =0 positions.1 of the bicycle! =0 moving.1 in the opposite directions. The# =0 cusps.2 if the rear track occur when the steering angle reaches 90◦ (not recommended in real life!)

These figures are taken from F. Wegner’s web site [25] called “Three problems – one solution”. The two problems are already mentioned, the flotation in equilibrium and bicycle track problem. The third is the motion of electron in the magnetic field whose strength depends quadratically on Fig. 13 m/n =1/4, Fig.∗ 14 the distanceFig. 15 m/n from=1 the/4 origin;, Wegner discovered that this problem is closely # =0.1 m/n =1/4, # =0.1 related# =0 to.2 the previous two. The site [25] contains an outline of Wegner’s work on these problems and numerous animations; see also [20]–[24]. Fig.∗ 19 Fig. 20 m/n =1/6, Fig.∗ 21 Based on the recent paper [3], we can add a fourth problem, that is closely m/n =1/5, ! =031.2 ! =0.05 m/n =1/6, ! =0.05 related to the above three. It is a variational problem to find closed planar curves that are relative extrema of the bending energy k2ds, where k is the curvature, with the length and area constraint; such curves are called buckled rings [9, 13]. One of the results of [3] is that theR solutions found by Wegner are buckled rings. One can say more: these curves are of a completely integrable system, the planar filament equationγ ˙ = k2 T + k0N that describes time 31 2 7 Fig.∗ 22 Fig. 23 m/n =1/7, m/n =1/6, ! =0.05 ! =0.1

7.2 Periodicity

In eq. (115) an angle of periodicity ψc has been defined. Here the periodicity is discussed for several regions in fig. 4. The angle of periodicity ψper is defined as the change of the angle ψ, as one moves from a point of extremal radius ri along the curve until a point of this extremal radius is reached again. Its sign is defined by the requirement that watching from the origin one starts moving counterclockwise. This yields ∆ψ ψper = , (202) dψ sign ( du ) r=ri ! ∆ψ = ψ(u +2ω3!) ψ(u) (203) ! −

32 evolution of a plane curve γ; here T and N are the unit tangent and normal vectors to the curve, k is its curvature, and k0 is the derivative with respect to the length parameter. The planar filament equation is a close relative of more physically mean- ingful filament equation, also known as smoke ring or local induction equa- tion. The filament equation describes evolution of space curves: each point moves in the binormal direction with the speed equal to the curvature, and it is one of the most studied completely integrable systems of type. The filament equation is also related to tire track geometry, see [18]. I am sure that Stanislaw Ulam would be pleased that his problem has connections to many seemingly unrelated subjects, and I hope that the float- ing in equilibrium problem will be completely solved, in all dimensions, in foreseeable future.

References

[1] V. Arnold. Arnold’s problems. Springer-Verlag, Berlin; PHASIS, Moscow, 2004.

[2] H. Auerbach. Sur un probl`emede M. Ulam concernant l’`equilibre des corps flottants. Studia Math. 7 (1938), 121–142.

[3] G. Bor, M. Levi, R. Perline, S. Tabachnikov. Tire tracks and integrable curve evolution. Int. Math. Res. Notes, published online 2018 https: //doi.org/10.1093/imrn/rny087.

[4] J. Bracho, L. Montejano, D. Oliveros. A classification theorem for Zindler carrousels. J. Dynam. Control Systems 7 (2001), 367–384.

[5] J. Bracho, L Montejano, D. Oliveros. Carousels, Zindler curves and the floating body problem. Period. Math. Hungar. 49 (2004), 9–23.

[6] R. Duda. Pearls from a lost city. The Lvov school of mathematics. Amer- ican Math.l Society, Providence, RI, 2014.

[7] R. Foote, M. Levi, S. Tabachnikov. Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture. Amer. Math. Monthly 120 (2013), 199–216.

8 [8] R. Gardner. Geometric tomography. Second edition. Cambridge Univer- sity Press, New York, 2006.

[9] G. H. Halphen. La corbe ´elastiqueplane sous pression uniforme, Chap. V in Trait´edes fonctions elliptiques et de leurs applications. Deuxi`eme partie: Applications a la m´ecanique,a la physique, a la g´eod´esie,a la g´eom´etrieet au calcul int´egral.Gauthier-Villars et fils, , 1888.

[10] Jo. Konhauser, D. Velleman, S. Wagon. Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries. Math. Assoc. of America, 1996.

[11] M. Levi. Schr¨odinger’sequation and “bike tracks” – a connection. J. Geom. Phys. 115 (2017), 124–130.

[12] M. Levi and S. Tabachnikov. On bicycle tire tracks geometry, hatchet planimeter, Menzin’s conjecture, and oscillation of unicycle tracks. Ex- perimental Math. 18.2 (2009), 173–186.

[13] M. M. L´evy. M´emoire sur un nouveau cas int´egrable du probl´emede l´elastiqueet l’une de ses applications. Journal de Math. Pures et Ap- pliquees s´er.3 X (1884), 5–42.

[14] G. Minton. Knotted Field Lines and Computer-Assisted Proof. https://jointmathematicsmeetings.org/amsmtgs/2168_ abstracts/1106-65-2198.pdf.

[15] H. Steinhaus. Mathematical Snapshots. Oxford University Press, New York, N. Y., 1950.

[16] H. Steinhaus. One hundred problems in elementary mathematics. Basic Books, Inc., Publishers, New York 1964.

[17] S. Tabachnikov. Tire track geometry: variations on a theme. Israel J. Math. 151 (2006), 1–28.

[18] S. Tabachnikov. On the bicycle transformation and the filament equation: Results and conjectures. J. Geom. Phys., 115 (2017), 116–123.

[19] S. Ulam. Problems in modern mathematics. John Wiley & Sons, Inc., New York 1964.

9 [20] F. Wegner. Floating bodies of equilibrium. Stud. Appl. Math. 111 (2003), 167–183.

[21] F. Wegner. Floating bodies of equilibrium I. arXiv:/0203061.

[22] F. Wegner. Floating bodies of equilibrium II. arXiv:physics/0205059.

[23] F. Wegner. Floating bodies of equilibrium. Explicit solution. arXiv:physics/0603160.

[24] F. Wegner. Floating bodies of equilibrium in 2D, the tire track problem and electrons in a parabolic magnetic field. arXiv:physics/0701241.

[25] F. Wegner. Three problems – one solution. http://www.tphys. uni-heidelberg.de/~wegner/Fl2mvs/Movies.html#float.

S. Tabachnikov Department of Mathematics Pennsylvania State University University Park, PA 16802, USA

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