Geometric Tomography · 1999-04-22 · Geometric Tomography R. J. Gardner 422 N OTICES OF THE AMS V...

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Geometric Tomography R. J. Gardner 422 NOTICES OF THE AMS VOLUME 42, NUMBER 4 T omography, from the Greek τ ´ oμoς, a slice, is by now an established and ac- tive area of mathematics. The word is usually associated with computerized tomography, the dramatic applica- tions of which include the CAT scanner in med- icine. In such applications, information is col- lected about sections of a density distribution. By various techniques, this information is syn- thesized to yield a reconstruction of the density distribution itself. For example, the CAT scan- ner produces images of two-dimensional sec- tions of human patients from X rays taken in a finite number of directions. Such a reconstruc- tion is always merely approxi- mate, the accuracy depending on the number of X rays, since no finite set of X rays deter- mines a density distribution uniquely. At the Conference on To- mography at Oberwolfach in 1990, the author introduced the term geometric tomography. In the author’s book [21], the fol- lowing definition is offered: Geometric tomography is the area of math- ematics dealing with the retrieval of information about a geometric object from data about its sec- tions, or projections, or both.” The use of the word geometric here is deliberately vague. A special case would be the study of sections or projections of a convex body or polytope, but it is sometimes more appropriate to consider star- shaped bodies, compact sets, or even Borel sets. . . . from the Greek τ ´ oμoς, a slice. The word projection is used in the sense of a shadow, that is, the usual orthogonal projec- tion on a line or plane. By considering only a strict subclass of den- sity distributions, one can sometimes obtain uniqueness in the inverse problem of deter- mining a set from partial knowledge of its sec- tions. For example, the author and McMullen [23] proved that there are certain prescribed sets of four directions in n-dimensional Euclid- ean space E n , such that the X rays of a convex body in these directions distinguish it from all other convex bodies. (In this context, an X ray gives the lengths of all chords of the body par- allel to the direction of the X ray; see Figure 1.) In such situations one might formulate and im- plement an algorithm providing an arbitrarily ac- curate reconstruction from a fixed finite set of X rays. Even when the collected data do not force a unique solution, interesting questions of prac- tical importance can be raised. For instance, what is the best way to estimate the volume of a three-dimensional convex body, given the areas of its projections on planes? Lutwak [34] ob- tained a striking upper bound by applying the Petty projection inequality, one of a slew of deep affine isoperimetric inequalities surveyed in [37]. Lutwak’s bound, which extends to E n , is indeed affine invariant—it yields the exact volume for ellipsoids! R. J. Gardner is professor of mathematics at Western Washington University in Bellingham, WA. His e-mail address is [email protected]. 1. Introduction

Transcript of Geometric Tomography · 1999-04-22 · Geometric Tomography R. J. Gardner 422 N OTICES OF THE AMS V...

Page 1: Geometric Tomography · 1999-04-22 · Geometric Tomography R. J. Gardner 422 N OTICES OF THE AMS V OLUME 42, NUMBER 4 T omography, from the Greek ˝´o o& , a slice, is by now an

GeometricTomographyR. J. Gardner

422 NOTICES OF THE AMS VOLUME 42, NUMBER 4

Tomography, from the Greek τoµoς, aslice, is by now an established and ac-tive area of mathematics. The word isusually associated with computerizedtomography, the dramatic applica-

tions of which include the CAT scanner in med-icine. In such applications, information is col-lected about sections of a density distribution.By various techniques, this information is syn-thesized to yield a reconstruction of the densitydistribution itself. For example, the CAT scan-ner produces images of two-dimensional sec-

tions of human patients from Xrays taken in a finite number ofdirections. Such a reconstruc-tion is always merely approxi-mate, the accuracy dependingon the number of X rays, sinceno finite set of X rays deter-mines a density distributionuniquely.

At the Conference on To-mography at Oberwolfach in1990, the author introduced theterm geometric tomography. Inthe author’s book [21], the fol-lowing definition is offered:

“Geometric tomography is the area of math-ematics dealing with the retrieval of informationabout a geometric object from data about its sec-tions, or projections, or both.” The use of theword geometric here is deliberately vague. Aspecial case would be the study of sections orprojections of a convex body or polytope, but itis sometimes more appropriate to consider star-shaped bodies, compact sets, or even Borel sets.

. . . from theGreek τoµoς,

a slice.

The word projection is used in the sense of ashadow, that is, the usual orthogonal projec-tion on a line or plane.

By considering only a strict subclass of den-sity distributions, one can sometimes obtainuniqueness in the inverse problem of deter-mining a set from partial knowledge of its sec-tions. For example, the author and McMullen[23] proved that there are certain prescribedsets of four directions in n-dimensional Euclid-ean space En, such that the X rays of a convexbody in these directions distinguish it from allother convex bodies. (In this context, an X raygives the lengths of all chords of the body par-allel to the direction of the X ray; see Figure 1.)In such situations one might formulate and im-plement an algorithm providing an arbitrarily ac-curate reconstruction from a fixed finite set ofX rays.

Even when the collected data do not force aunique solution, interesting questions of prac-tical importance can be raised. For instance,what is the best way to estimate the volume ofa three-dimensional convex body, given the areasof its projections on planes? Lutwak [34] ob-tained a striking upper bound by applying thePetty projection inequality, one of a slew of deepaffine isoperimetric inequalities surveyed in [37].Lutwak’s bound, which extends to En, is indeedaffine invariant—it yields the exact volume forellipsoids!

R. J. Gardner is professor of mathematics at WesternWashington University in Bellingham, WA. His e-mailaddress is [email protected].

1. Introduction

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Much of geometric tomography, as expoundedin [21], consists of inverse problems concerningdata of one of three types: X rays, measurementsof projections, and measurements of concur-rent sections. We shall explain these terms andprovide just a sampler of the many results ineach category, focusing on uniqueness and ig-noring other types of data and intriguing topicssuch as stability, reconstruction, and estimatesof volume. We stress that while the current stateof the art often makes convexity a convenient as-sumption, this is usually unnecessary, exceptwhere projections are involved, and sometimeseven inappropriate.

2. X RaysIn 1917, J. Radon showed that a density distri-bution in the plane is determined by its X raystaken in every direction. To prove this, Radonfound an inversion formula for an integral trans-form now called the Radon transform; the for-mula is a key ingredient in computerized to-mography, though much more is required inpractice (see [42]).

Geometric tomography is concerned with sets,that is, densities taking only the values 0 or 1.Suppose that C is a compact subset of En andu is a direction, a unit vector in the unit n-sphereSn−1. Denote by u⊥ the (n− 1)-dimensional sub-space orthogonal to u. The parallel X ray of Cin the direction u gives for each x ∈ u⊥ the lin-ear measure of the intersection of C with the linethrough x parallel to u. A slight strengtheningof Radon’s theorem implies that if U is an infi-nite set of directions in S1, then the parallel Xrays of a compact set in E2 in the directions inU distinguish it from any other compact set.

In 1963, P. C. Hammer posed several problemsconcerning the determination of planar convexbodies by finite sets of X rays. Here there are sim-ple examples of nonuniqueness: A regular n-gon P and its rotation Q by π/n about its cen-ter have equal parallel X rays in each of the di-rections of the edges of the regular 2n-gon Rformed by the convex hull of P and Q . Moreover,since affine maps preserve the ratios of lengthsof parallel line segments, the images φP and φQof these n-gons under an affine map φ haveequal parallel X rays in each of the directions ofthe edges of the affinely regular polygon φR.The author and McMullen [23] proved that theseare the only sets of directions to be avoided inorder to obtain uniqueness. To be precise: If Uis a finite set of directions in S1, then the cor-responding parallel X rays of a planar convexbody distinguish it from any other such body ifand only if U is not a subset of the directions ofthe edges of an affinely regular polygon. In con-trast to Radon’s theorem, the proof does not useintegral transforms, though these also repre-

sent a major tool in the geometric branch of to-mography.

To obtain the result mentioned in Section 1,one observes that the cross ratio of the slopesof any four edges of a regular polygon is an al-gebraic number. This remains true of an affinelyregular polygon, since affine maps preserve

cross ratio. Therefore a set of four directionswhose slopes have a transcendental cross ratiowill ensure that the corresponding parallel Xrays determine each planar convex body. Con-vex bodies in En can also be determined by par-allel X rays in such a set of four directions lyingin the same 2-dimensional plane, since the Xrays then determine each 2-dimensional sliceparallel to this plane.

One can also imagine an X ray in which thebeams are not parallel but rather emanate froma single point; in fact, such “fan-beam X rays” areemployed in modern CAT scanners. Mathemat-ically, the point X ray of C at a point p in En givesfor each u ∈ Sn−1 the linear measure of the in-tersection of C with the line through p parallelto u.

The problem of how best to determine con-vex bodies by point X rays has not been com-pletely solved. One point X ray is clearly insuf-ficient; see Figure 2. There are results concerningX rays at two points—Falconer [15] found aclever way of applying a version of the stablemanifold theorem—but these require advanceknowledge of the position of the body relativeto the two points. The best unrestricted unique-ness theorem was established by Volcic [49]:Every set of four noncollinear points in the planehas the property that X rays of a planar convexbody at these points distinguish it from anyother such body.

C

u

Graph of X-ray

u⊥

o

Figure 1A parallel X ray

of a convex body C.

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Many well-known problems fit squarely intothe framework of geometric tomography. Forexample, the notorious equichordal problem canbe phrased as follows: Is there a planar body,bounded by a simple closed curve and starshaped with respect to two interior points pand q, whose point X rays at p and q are bothconstant? Posed in 1917, the equichordal prob-lem generated deep studies by E. Wirsing,R. Schäfke, H. Volkmer and others. Now theproblem has apparently been laid to rest: a tourde force by Rychlik [44] offers a negative solu-tion.

3. ProjectionsIf the compact set C is a nonempty body—thatis, C is the closure of its interior—then the sup-port of the X ray of C in a direction u is just theprojection C|u⊥ of C on u⊥, so the latter yieldsstrictly weaker information. This link is one rea-son for allowing projections as well as sectionsin geometric tomography. (Polar duality pro-vides another; see below.)

Holes and dents can be invisible in shadows,so the class of convex bodies is the largest con-venient class for this type of data. Let us con-sider two basic measurements of their projec-tions. The width function of a convex body Kgives for each u ∈ Sn−1 the length of the pro-jection (of K) on the line through the origin par-allel to u. The brightness function of K gives foreach u ∈ Sn−1 the volume of its projection onu⊥. (The volume of a k-dimensional body al-ways means its k-dimensional volume.) In E3,then, the brightness function of a convex bodygives the areas of its shadows cast on planes.

Mixed volumes furnish an ideal vehicle forproblems involving projections as well as a uni-fied treatment of metric quantities such as vol-ume, surface area, and mean width. They play a

central role in the Brunn-Minkowski theory,founded by H. Minkowski in the last decade ofthe nineteenth century after groundwork ofJ. Steiner and H. Brunn. (Schneider’s book [46]is a lucid and comprehensive guide to thislabyrinthine palace, and references to every-thing in this section are supplied there.)Minkowski’s discoveries included generaliza-tions of the classic isoperimetric inequality andemployed basic tools of analysis such as mea-sure, integral, and spherical harmonics. TheBrunn-Minkowski theory quickly produceddozens of theorems on the important sets of con-stant width or brightness (the front cover pic-ture depicts one of the latter) and characteriza-tions of spheres or ellipsoids in terms of variousmeasurements of their projections.

Perhaps the single most important develop-ment from the point of view of geometric to-mography came with a uniqueness theorem ofA. D. Aleksandrov. In 1937, he (and, indepen-dently, W. Fenchel and B. Jessen) introduced thesurface area measure of a convex body, a con-cept allowing many results in convex geometryto be relieved of unnecessary differentiability as-sumptions. This innovation is employed in Alek-sandrov’s uniqueness theorem for projections:A convex body in En that is centered, centrallysymmetric with center at the origin, is deter-mined, among all such bodies, by its brightnessfunction. The proof hinges on two facts: An in-tegral transform called the cosine transform isinjective on the even continuous functions onSn−1, and a convex body is determined up totranslation by its surface area measure. The lat-ter is a consequence of the Aleksandrov-Fenchelinequality, a profound generalization of theisoperimetric inequality that has unexpectedconnections with other areas, for example, al-gebraic geometry.

In the later development of the Brunn-Minkowski theory, the concept of a projectionbody is of special significance. This is a cen-tered convex body whose support function—giving for each u ∈ Sn−1 the distance from theorigin to a supporting hyperplane orthogonal tou—is the brightness function of another convexbody. Figure 3 depicts the projection bodies ofa regular tetrahedron and double cone. To il-lustrate the role of projection bodies, supposemeasurements of a centered convex body in En,n ≥ 3, yield the comparative information that itsbrightness function is smaller than that of an-other such body. Can one conclude that its vol-ume is also smaller? Sometimes called Shep-hard’s problem, this was solved in 1967 byC. M. Petty and R. Schneider; the answer is yesif the body with the larger brightness functionis a projection body, but no in general for each n.

C

C' C'

C

p

p

Figure 2Polygons withequal point X rays at p.

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4. Concurrent SectionsIn geometric tomography there is a remarkable andmysterious correspondence between projectionsand concurrent sections, sections through a fixedpoint that can be taken as the origin. Some con-nection between the two can be expected from thepolar duality familiar to all geometers. If K is aconvex body containing the origin in its interior,denote the polar body of K by K∗; (see [46], p.33). (If K is centered, its support function, ex-tended from Sn−1 to a positively homogeneousfunction on En, is a norm on En. Then K∗ is theunit ball in the corresponding normed space(En,K∗). This is the source of much interplay be-tween convexity and functional analysis.) If S isa subspace, then (see [38], p. 70) it is easilyshown that K∗ ∩ S = (K|S)∗, where the polar op-eration on the right is taken in S . Apart from afew special situations, however, this equationsheds little light on the puzzle.

If, in the definitions above, we replace thewords “projection on” with “section by”, thewidth function of a convex body becomes itspoint X ray at the origin and the brightness func-tion becomes a new function called the sectionfunction that gives for each u ∈ Sn−1 the volumeof the intersection with u⊥. When the measureddata concern planar sections through the origin,however, it is more appropriate to work with starbodies than with convex bodies. A star body isa body, containing the origin and star shapedwith respect to the origin, whose radial func-tion—giving for each u ∈ Sn−1 the distance fromthe origin to the boundary in the direction u—is continuous. (For another, more general defi-nition, see [24].) The definitions of point X rayat the origin and section function extend natu-rally to star bodies.

We have mentioned one of the founders of theanalytical basis of tomography, J. Radon. An-other was P. Funk, whose paper [17] proves: Acentered star body in E3 is determined, amongall such bodies, by its section function. Refor-mulated, Funk’s uniqueness theorem says thatan integral transform called the spherical Radontransform is injective on even continuous func-tions on Sn−1. The latter was actually proved ear-lier by Minkowski for n = 3; subsequently, proofsfor general n were found by H. Helgason, C. M.Petty, and R. Schneider.

Though perfect for projections, the Brunn-Minkowski theory has comparatively little rele-vance for sections. In 1975, Lutwak [33] initiateda “dual Brunn-Minkowski theory”, in which pro-jections of convex bodies are replaced by sec-tions of star bodies. He observed that the inte-grals over Sn−1 of certain expressions involvingpowers of radial functions behave in many waysjust like mixed volumes, and called them dualmixed volumes. Moreover Hölder’s inequality,

when reformulated in terms of dual mixed vol-umes, looks just like the Aleksandrov-Fenchel in-equality, with the inequality reversed!

In these early ingredients, the dual theory ismore Bauhaus than Byzantine, though no less ef-fective. But Lutwak’s dual theory includes muchmore than this. In [35], for example, he intro-duced the notion of an intersection body, a cen-tered star body whose radial function is the sec-tion function of another star body. Theintersection body of a convex body need not beconvex, but a theorem of H. Busemann, funda-mental for his theory of area in Finsler spaces,can be reinterpreted as saying that the inter-section body of a centered convex body is con-vex. Some examples of convex intersection bod-ies are illustrated in Figure 4. On the left, acentered cube C of side length 1/2 includes itsintersection body, created from a program writ-ten by Fred Pickle; on the right, a centered cylin-

der K of radius 1 and height 2 is shown withits (translated) intersection body.

The Brunn-Minkowski theory and Lutwak’sdual theory feed from each other, with greatbenefit for both and for geometric tomography.This process occurs via a sort of dictionary thatallows transport from one theory to the other.In this dictionary, convex bodies, projections, thesupport function, the brightness function, mixedvolumes, the cosine transform, and projectionbodies in the Brunn-Minkowski theory corre-spond to star bodies, sections by planes throughthe origin, the radial function, the section func-tion, dual mixed volumes, the spherical Radontransform, and intersection bodies, respectively,in the dual theory. In this way, the uniquenesstheorems of Aleksandrov and Funk, for exam-ple, translate into each other. Moreover, a proof

T C ΠCΠT

Figure 3Projection

bodies.

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in one theory can sometimes be translated intoa proof in the other.

When we translate Shephard’s problem ac-cording to Lutwak’s dictionary, we obtain the fol-lowing problem: If the section function of a cen-tered star body in En, n ≥ 3, is smaller thanthat of another such body, is its volume alsosmaller? The answer is yes if the body with thesmaller section function is an intersection body,but is in general no for each n. (The word“smaller” is due to the inequality reversal men-tioned above.)

The first part of this result is a theorem of Lut-wak [35]. Lutwak was actually motivated by avariant of this problem, obtained by replacingthe word “star” in the previous paragraph by“convex”. This is precisely the well-known Buse-mann-Petty problem, one of a list of ten prob-lems in [10] inspired by investigations inMinkowskian geometry. Lutwak’s theorem stillapplies, of course, and intersection bodies—and,by the way, nonconvex star bodies—also play acrucial role in the solution to the Busemann-Petty problem. In fact, the answer to the Buse-mann-Petty problem is yes for a given n if andonly if each sufficiently smooth centered convexbody in En is an intersection body. This discov-ery led the author [18,19,20] and Zhang [52,53]to the complete solution; it turns out that the an-swer is no for each n ≥ 4 but an unqualified yesif n = 3. Along the way, Ball [1] established theprecise upper bound of

√2 for the section func-

tion of a centered cube of unit side length. Whileexpected, the result is tricky to prove, and, in-cidentally, yields a negative answer to the Buse-mann-Petty problem for n ≥ 10.

Recent work of Goodey, Lutwak, and Weil [25]suggests that a profound synthesis of the Brunn-Minkowski theory and its dual may one day berevealed, and the mystery explained at last.

5. Public Relations DepartmentComputerized tomography generally deals withsections of density distributions. Despite this, itis worth noting that geometric objects do occurin the medical literature. For example, the au-thors of [11] approximate the human heart by aconvex body and attempt to use two X rays toobtain a reconstructed image. However, appli-cations of geometric tomography are perhapsmore likely to occur in several other areas.

Skiena [48] envisages that what he calls “geo-metric probing” will be of use in robotics. Arobot may need to be equipped with a sensingdevice to help identify the shape and positionof geometric objects, for example, in pickingmachine parts off a conveyor belt or in movingaround an environment in which objects are ap-proximated by polyhedra. Skiena investigatesthe possibility of using X rays, and this leads him

to consider X rays of (convex or nonconvex)polytopes. The fresh “interactive” viewpoint ofcomputer science brought the natural and prac-tical concept of successive determination, inwhich X rays already taken can be consulted indeciding the direction for the next X ray; see [14]and [22]. X rays of geometric objects also occurin Horn’s book [29] on robot vision.

On September 19, 1994, a minisymposiumwith the title “Discrete Tomography”, organizedby Larry Shepp of AT&T Bell Labs, was held atDIMACS. Some time earlier, Peter Schwander, aphysicist at AT&T Bell Labs in Holmdel, hadasked Shepp for help in obtaining three-dimen-sional information at the atomic level from two-dimensional images taken by an electron mi-croscope. A new technique, based onhigh-resolution transmission electron mi-croscopy, can effectively measure the number ofatoms lying on each line in certain directions. (Atpresent, this can be achieved only for some crys-tals and in a constrained set of crystallographicdirections—lattice directions for the crystal lat-tice.) The aim is to determine the three-dimen-sional crystal from information of this sort ob-tained from a number of different directions.This leads to the problem of determining a fi-nite set from its projections (counted with mul-tiplicity) on a finite number of planes. While itformally belongs to geometric tomography, theproblem is given only very brief mention in [21];partial answers can be found in [7] and [16].

Suppose that C is a compact set in En and Sis a k-dimensional subspace, where 1 ≤ k ≤n− 1. The k-dimensional X ray of C parallel toS gives for each x ∈ S⊥ the volume of the in-tersection of C with the translate of S contain-ing x. The ordinary X ray corresponds to k = 1.In [12], it is observed that the problem of radartarget estimation is related to the determinationof a three-dimensional body from its two-di-mensional X rays. The k-dimensional X ray of aconvex polytope P, parallel to a subspace ingeneral position with respect to the vertices ofP, is a piecewise polynomial of degree at mostk, at least (k− 1)-times continuously differen-tiable. In short, it is a spline. Of particular con-sequence in spline theory are the cases when Pis a simplex or a cube, the latter giving rise tobox splines. Box splines are of considerable in-terest in computer-aided design; see [13].

If C is a body and u is a direction, the maxi-mum volume of sections of C by hyperplanesorthogonal to u is sometimes called an HA mea-surement. This strange term derives from thestudy of the Fermi surface of a metal—the sur-face of a body formed, in velocity space, by ve-locity states occupied at absolute zero by valenceelectrons of the metal. Knowledge of the Fermisurface gives valuable information about various

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physical properties such as conductivity. Themaximal cross-sectional areas of the bodybounded by a Fermi surface can be measured bymeans of the de Haas-van Alphen effect, mag-netism induced in the metal by a strong magneticfield at a low temperature, hence the name HAmeasurement. More details and references areprovided by Klee [32].

Projection bodies appear in quite surprisingguises. A strong form of Liapounov’s theoremsays that they are precisely the ranges of vector-valued measures. A centered convex polyhedronis equidissectable with a cube if and only if it isa projection body. If K is a centeredconvex body in En , the normedspace (En,K∗) defined above is iso-metric to a subspace of L1(0,1) ifand only if the body is a projectionbody. Projection bodies have alsofound application in stochasticgeometry, random determinants,Hilbert’s fourth problem, math-ematical economics, and other areas;see [26,36], and the references giventhere.

Santaló ([45], p. 282) quotes a de-finition of stereology, due to H. Elias,as the exploration of three-dimen-sional space from two-dimensionalsections or projections of solid bod-ies. The relatively new term wascoined at a meeting, principally of bi-ologists and metallurgists, on theFeldberg, Germany, in 1961. Stere-ology is closely related to geometrictomography, but focuses on statis-tical estimates from random sec-tions. Many pertinent references can be foundin [45] and in Weil’s survey [50]. Mathematicalmorphology, image analysis, and pattern recog-nition also overlap with geometric tomography.

6. Try Your Hand?Space permits only one open problem from eachof the three categories referred to at the end ofSection 1. They all happen to concern convexbodies, but many of the seventy or so open ques-tions listed in [21] do not.

Question 1. In E3 , is there a prescribed finite setof directions in general position such that the Xrays of any convex body in these directions willdistinguish it from any other such body?

Here, general position means that no three ofthe directions lie in a plane. The four directionsin the theorem of the author and McMullen [23](see Sections 1 and 2) all lie in the same plane.Unfortunately, an arbitrarily small perturbationof a suitable set of four directions can destroythe uniqueness property, so that the theorem

lacks stability. Question 1 is especially interest-ing because it seems possible that any set ofseven (mutually nonparallel) directions in gen-eral position will do; an example found by theauthor, Volcic , and Wills ([21], Theorem 2.2.2)shows that not every set of six directions in gen-eral position has the required property.

Question 2. Is there a nonspherical convex bodyin En, n ≥ 3, of constant width and constantbrightness?

This tough old nut goes back to Nakajima[41], who showed that the answer is no for n = 3

under an additional smoothness assumption.Nonspherical convex bodies of constant widthhave been around since Euler named them orb-iforms, and Blaschke ([8], pp. 151–4) constructednonspherical convex bodies whose brightnessfunctions are constant. Centered balls are theonly star bodies whose point X rays at the ori-gin and section functions are both constant; thisanswer to the dual form of Question 2 is givenin [24] (see also [27]).

Question 3. Is there a constant c, independentof n, such that if H and K are centered convexbodies in En with the section function of H smallerthan that of K, then the volume of H is smallerthan c times the volume of K?

A complete answer to a problem on Buse-mann and Petty’s list ([10], Problem 2) would alsoanswer Question 3; but, as formulated here, withthe emphasis on the universal constant c, thequestion is a quite recent one. Bourgain [9]proved that c can be replaced by cn = O(n1/4).The solution to the Busemann-Petty problem

K IKC

IC

Figure 4Intersection

bodies.

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(see Section 4) shows that c3 = 1 is possible, butcn > 1 for n ≥ 4. Sometimes called the hyper-plane problem or the slicing problem, Question3 is also known as the maximal slice problem,in view of one of several equivalent forms ex-amined at length by Ball [2] and Milman andPajor [39]. Contributions to the slicing problem,which is of great significance in the local theoryof Banach spaces, have been made by Ball [3,4],Junge [30,31], and Zhang [54].

If we replace the section function in Question3 with the brightness function, the answer is no.Ball [5,6] shows that in this case c can be replacedby cn = O(n1/2) and that this order is the correctone, even for arbitrary convex bodies.

7. PostscriptThere must be thousands of mathematical re-sults involving sections or projections of com-pact sets. Inevitably, several worthy topics wereomitted. For example, Dvoretzky’s theorem saysthat given k ∈ N , each centered convex body ofsufficiently high dimension has an “almost spher-ical” k-dimensional central section. This seededa whole branch of Banach space theory concen-trating on the properties of high-dimensionalconvex bodies, expounded, for example, in thebooks of Milman and Schechtman [40] and Pisier[43]. Another example is Crofton’s intersectionformula and others from integral geometry, forwhich the texts of Schneider ([46], Chapter 4) orSchneider and Weil [47] can be recommended.

The scope can also be widened in other ways.There are already some results on X rays in pro-jective space, and several of the concepts dis-cussed above carry over to the more general ho-mogeneous spaces, as in the work of Helgason[28] and others. In fact, readers should interpretor even broaden the definition of geometric to-mography above to suit their tastes. Fascinatingnew possibilities arise, for example, in inverseproblems involving intersections with circlesand spheres, treated in the eloquent article byZalcman [51].

A definition should not sever new shoots ofmathematics, nor shade its beauty.

References1. K. M. Ball, Cube slicing in Rn, Proc. Amer.

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