Infinitesimal Aspects of the Busemann-Petty Problem

INFINITESIMAL ASPECTS OF THE BUSEMANN-PETTYPROBLEM

ERIC L. GRINBERG AND IGOR RIVIN

0. Introduction

In an old paper [6] H. Busemann and C. Petty pose the following problem. Doesthere exist a pair of convex, origin-symmetric bodies Bx and B2 in Rn so that for everyhyperplane H through the origin we have

while

We shall refer to this phenomenon as inequality reversal. There has been considerableprogress made in understanding this problem, but it remains open in its fullgenerality. The hypotheses of convexity and origin-centredness above were imposedafter Busemann [5] found fairly simple counterexamples in cases where at least oneof these conditions is removed. Below we exhibit even simpler examples.

The initial motivation for this problem probably came from an old inequality ofBusemann [4]:

Vol^M)""1. (1)HettP"

Here the integration is over the set RPn~x of hyperplanes H through the origin inRn, dH is the normalized rotation-invariant measure on RPn~\ Voln_x(M n H)denotes the cross-sectional area of the body M in H, and Voln(M) is the Euclideanvolume of M. The constant cn is chosen so that (1) is an equality if M is the unit ballcentred at the origin. The inequality is valid for M a measurable body in Rn

(Busemann proved (1) for convex bodies; the extension to the measurable categorywas given by Petty in [13]). The «th root of the left-hand side of (1) is called the 1stdual affine Quermassintegral of the body M and is denoted by OX(M) (see Lutwak [11,12]). Busemann showed that if equality holds in (1) and M is convex, then M mustbe an origin-centred ellipsoid. A ^-dimensional version of the equality portion of (1)was given by Furstenberg and Tzkoni [7] for M an origin-centred ellipsoid:

(M n H)}n dH = Voln(Mf.HeG(k,n)

Here the integration is over G(k, n), the Grassmann manifold of ^-dimensionalvector subspaces of Rn. The ^-dimensional version of the mequality along with a

Received 17 April 1989; revised 16 March 1990.

1980 Mathematics Subject Classification 52A40, 53C65.

First author supported in part by a grant from the National Science Foundation; second authorsupported by a contract from the Defense Advanced Research Projects Agency.

Bull. London Math. Soc. 22 (1990) 478-484

INFINITESIMAL ASPECTS OF THE BUSEMANN-PETTY PROBLEM 479

complex version were proved recently by Grinberg [9]. It is immediate from (1) thatinequality reversal cannot occur in the Busemann-Petty problem if B2 is an origin-centred ellipsoid: for B2 the Euclidean volume is precisely {Q>1(B2)}

ntln~1). But this isstrictly smaller than {<t>1(B1)}

n/(n~1) which, by (1), is no larger than the volume of Bv

With a slightly weaker convexity assumption, Giertz [8] showed that in R3 if both Bx

and B2 are surfaces of revolution (about the same axis) then inequality reversal cannotoccur. Since the inequality (1) goes only one way, it is not excluded that there existsan example of inequality reversal with Bx an origin-centred ellipsoid. Surprisingly,Larman and Rogers [10] proved the existence of just such an example. They work indimensions higher than 11, but their methods may work in lower dimensions as well.The technique gives an existence theorem, but no specific examples. Perhaps evenmore surprisingly, it was observed recently that if n ^ 9 then inequality reversaloccurs with B2 the unit origin-centred cube and Bx an origin-centred ball ofappropriate radius. (This folk-theorem was discovered by a number of people afterK. Ball [1] found the largest hyperplane cross-sections of the «-cube; see Ball [2].)E. Lutwak [11, 12] has shown recently that inequality reversal cannot occur for asubstantial class of bodies. Still, it remains to decide in which dimensions inequalityreversal can occur, and for which bodies. In this paper we shall discuss a variation ofthe Busemann-Petty problem. Namely, we shall examine the possibility of deformingthe body B through a 1-parameter family of bodies {Bt}t>0 so as to reverse inequalitiesfor t small. We shall exhibit a class of bodies that includes origin-centred ellipsoidsfor which infinitesimal inequality reversal is not possible. We are indebted to thereferee for a careful reading of the manuscript and for some useful suggestions andreferences.

1. Some counterexamples

Let B be an origin-symmetric star-shaped body in R". For each non-zero vectorxeR" we denote by pB(x) the chord length Vol^i? n [JC]), where [x] is the linedetermined by the vector x. The function pB(x) is an even function on the sphere S1""1

(equivalently, a function on RP""1) called the radial function of the body B. Clearly,B is determined by its radial function. By integrating in polar coordinates we see that

{pB[x]}nd[x].i-i

In this section, cn will stand for a positive constant (depending on the normalizationof some measures) that may vary from equation to equation and whose exact valuemay be determined by substituting the unit ball for the convex body B. If H is ahyperplane through the origin in R" and [H] is the corresponding projectivehyperplane, then

Instead of looking for bodies B we shall search for radial functions f[x] on RP""1.Since we shall be dealing with families of bodies B, we introduce a function ft[x] onRP""1 parametrized by the real variable t. For simplicity, the functions/consideredwill be of the form

ft[x)=f[x]

480 ERIC L. GRINBERG AND IGOR RIVIN

Thus we can think of/[x] as the base function and g[x] as the perturbation function.A differentiation under the integral sign shows that

a\Oln(Bt) , M/-n-ir~i ~r~i jr«i ^2)- It-0 J [3

'[zJeRP"

anddYo\n(Bt n H)

dt= cn\ (n-\)fn-2[x]g[x]d[x]H. (3)

t-0 J[x]e[H]

If the function/is to represent a body for t small, then/[x] should be a strictly positivefunction (so that B will have positive dual affine Quermassintegrals). The functiong[x] is unrestricted. In light of equations (2) and (3), in order to produce an inequalityreversing family ft[x] it suffices to find a positive function/[jc] and a function g[x] (bothpresumably measurable or continuous) so that /RP"-'/""1^ is positive, while for anyhyperplane [H], j[H]f

n~2g is negative (or vice versa). Clearly, this cannot be done if/n~x =/n~2, that is, if/[jc] essentially takes on only the values 0 and 1. In particular,this holds fo r /= 1, that is, for the unit ball. This leads to the following.

PROPOSITION. // is not possible to deform an origin-centred ellipsoid so as toinfinitesimally reverse the Busemann-Petty inequalities.

To turn the above remarks into a proof, one first observes that the problem isinvariant under the general linear group in the following sense: a lineartransformation Te GL (n, R) multiplies the volume element in Rn by a constant andsends the Euclidean area element on a plane H to a constant multiple of the Euclideanarea element on T(H). Hence an inequality reversing deformation can be performedon a given convex body B if and only if it can be so done on any of the affine(GL («)) images of B. This reduces the case of an ellipsoid to the case of a ball.Next, we considered above families of the special form f[x] +tg[x]. But a standardapplication of Taylor's theorem (or the Malgrange preparation theorem) shows thatthere is no loss of generality.

To construct an example of inequality reversal, we remove the convexityassumption and work in R3. Our base body will be constructed by tightening a beltaround an origin-centred ball. Since the ball is given by a constant radial function, wecan describe the belt-tightened body by

(1, outside the belt,

y, inside the belt.

Here y is a positive number which will be fixed later. The deformation of our basebody will be achieved using the perturbation function

(K, outside the belt,g = {

[ — 1, inside the belt.

Here K is a positive number to be fixed later. The value — 1 inside the belt indicatesfurther belt tightening. As for the belt itself, we shall take it to be an equatorial beltof longitudinal angular extent e. Although the functions/and g are not continuous,


they are closable, so that the resulting (bodies are solid connected objects. If B is thebody with radial function / + tg (t > 0), then in order to have inequality reversal for/ near 0 we desire that

\ fg<0, (4)J[H]

for every projective hyperplane [H] in RP2, or equivalently, every great circle H in theorigin-centred unit-radius sphere S2. Equation (4) says that for small t, the cross-sectional area Vol2(2?f n H) is smaller than Vol2(l?0 n H). The worst-case scenariooccurs when the great circle H intersects the belt orthogonally (g is negative on aslarge a set as possible). For the specific form of ft above, the variation in the areaintegral in question is K(\— e)n — eyn, where the first term represents the integraloutside the belt and the second term represents the integral inside the belt. This isbecause the integrand fg takes on the value K on the arc of length (1 - e) n in H outsidethe belt, and takes on the value - y on the arc of length en in H inside the belt.Similarly, the variation in volume is given by (2 — e)Kn — yhn. Thus, to achieveinequality reversal we need to find positive constants K, g and e (with K and g near1, e small) so that

K(l-e)n-eyn < 0,

(2-8)Kn-y2en>0.

This system of inequalities is easy to solve. For instance, we can take K = 1, £ = £ andy = 1.1. Thus to achieve the inequality reversal we start with a ball whose belt hasbeen tightened, and then tighten the belt further while at the same time bloating thecomplementary region.

We now exhibit inequality reversal using non-origin-centred bodies. The basicidea is to 'just take a body and move it away from the origin'. In detail, let B be theunit ball in Rn and consider the rigid motion M(t):x\-+x + tv, where v is a unitvector. Put Mt{B) = {M{i)b\beB}. For each fixed hyperplane H through the origin,the area a(i) = aH(t) = Vol^dM/i?)} n H) is a non-increasing function of t. Initially,u(t) is strictly decreasing in /, unless H = (v)1. Since rigid motions preserve volume,we almost have inequality reversal. To make all central cross-sections decrease,observe that for / large, aH{t) is zero unless H is close to (y)1. We now consider the

17 BLM 22


semi-simple linear transformation D(e) which dilates in the v direction by a factor of(1—e) and by n~1-\/\/(\—e) in the (v)1 directions. D(e) preserves volume, and (forsmall e)

n H}) < voucwtf)} n H),

for H sufficiently close to (v)x. Thus Bx = B has the same volume asD(e){(Mt(B)) n H}, but all its central cross-sections are larger. Letting B2 be a slightlyexpanded copy of D(e){(Mt(B)) 0 H}, we obtain inequality reversal. It is possible tosimply write down an equation for the body involved, but one hopes that the abovedescription is more illuminating.

2. Controlling inequality reversal

We now strive to prove that for a large class of bodies, infinitesimal inequalityreversal is not possible. We have identified a body B with an everywhere positivefunction/!*] on RP""1 and the inequality reversal condition involves integrals of/"~2gand/""1^, where g = g[x] is an unrestricted function on RP""1. Thus we can replacefn~2g by g and ask for which strictly positive function f[x] the condition

g<0)

(g a fixed function on RP""1 and H an arbitrary hyperplane in RP""1) implies

fg<0.RP"

We have observed that this implication is valid for /= 1. To extend the validity to awider range of functions, we consider a distribution (in the sense of L. Schwartz) first.Let y be a fixed point on RP""1 which we think of as the north pole on S""1. Later,we shall let y vary. The following functional is a distribution on C^RP""1) in thesense of L. Schwartz:

f g[x]dH[x]dH.{«|northpolee//} J [H]

The outer integral ranges over the hyperplanes H through the origin in Rn that meety, and dH is normalized rotation-invariant measure on this set. Note that {H\ northpolee//} is an equator, or a 'great' RPn~2 c RPn-1; the symbol dH used here is arestriction of the volume element dH used globally on RP""1 in Section 0. The innerintegral is over the projective («—l)-plane H with the standard measure. Thedistribution v is of order zero and so it can be realized by a function f[x]:

JRf[x]g[x]d[x].


For instance, if n = 3 and we take spherical coordinates (0,0, z), then an elementaryintegration shows that/(0, 6,z) is just l/sin(0). Clearly, it is not possible to reverseinequalities infinitesimally starting with a body corresponding to / . This is because

g < 0 for every projective hyperplane H

implies

fg<0,J R

as follows from the original distributional definition of / . The function / is notbounded, so it corresponds to an unbounded body B. In fact, B is an infinite cylindercentred about the axis defined by the origin and the north pole. On the other hand,if one attempts to perturb the belt-tightening example above to make the base bodyconvex, one arrives at the infinite cylinder, and here there is an obstacle. Thus, theconclusion that the infinite cylinder is, at least formally, non-inequality-reversiblesuggests that the same is true of a large class of convex bodies.

The infinite cylinder is unbounded and therefore is not in the class of bodies forwhich the original Busemann-Petty problem was posed. To obtain results aboutgenuine bodies we vary the north pole x in the above construction. We shall makeexplicit the dependence of the function / on the choice of north pole y by writingf = f[x] = /„[*]. Now choose a positive measurable function h[x] on RP""1. Put

h[y]fy[x]dy.l/eRP""1

We shall at times abuse notation and denote Fh[x] simply by F[x]. The functionF[x] is a positive superposition of functions fy[x], and therefore the correspondingbody B = Bh cannot be perturbed so as to infinitesimally reverse inequalities. Moreover,if the function h[x] is smooth, then, viewing the definition of Fh[x] as a convolution,it follows that F[x] is a smooth function and hence the body Bh is a smooth(in particular bounded) body. Let $} be the class of bodies

36 = {solid bodies B\ B = Bh for some positive function heL^RP11'1)}.

THEOREM. Let B be a body in Rn belonging to the class (%. Then it is not possibleto deform B and infinitesimally reverse the Busemann-Petty inequalities.

The class && contains a collection of bodies that is quite familiar to convexgeometers. Let K be a body in Rn (not necessarily convex). Let I(K) be the bodywhose radial function is given by

PiW[x] = VoUC* n [x]1) = f pT1 4 W (5)J[x]x

It was proved by Busemann [3] that if Kis convex then so is I(K). Bodies of the formI{K) are called intersection bodies. (The right-hand side of (5) makes sense for anymeasurable function p[x] and leads to the notion of generalized intersection bodies.)As a consequence of the theorem we have the following.

PROPOSITION. It is not possible to deform an intersection body so as to reverse theBusemann-Petty inequalities.


Actually, slightly greater generality is possible: one can introduce a positiveweight factor in the integral defining the infinite cylinder function / = / „ above. Itwould be interesting to determine exactly which bodies B belong to the class 08.

Unfortunately, the positivity assumption on h makes it rather difficult to settle thisquestion, but Bochner's criterion for the Fourier transform of a positive function mayhelp. Still, it is possible to generate many specific elements Be 08 by explicitlyconvolving some particular /is, say using spherical harmonic expansions.

References

1. K. BALL, 'Cube slicing in R"\ Proc. Amer. Math. Soc. 97 (1986) 465-473.2. K. BALL, 'Some remarks on the geometry of convex sets', Lecture Notes in Math. 1317 (Springer,

Berlin, 1988), pp. 224-231.3. H. BUSEMANN, 'A theorem on convex bodies of the Brunn-Minkowski type', Proc. Nat. Acad. Sci.

USA 38 (1949) 27-31.4. H. BUSEMANN, 'Volume in terms of concurrent cross sections', Pacific J. Math. 8 (1953) 1-12.5. H. BUSEMANN, 'Volumes and areas of cross-sections', Amer. Math. Monthly 67 (1960) 248-249,

corrections p. 671.6. H. BUSEMANN and C. PETTY, 'Problems on convex bodies', Math. Scand. 4 (1956) 88-94.7. H. FURSTENBERG and I. TZKONI, 'Spherical harmonics and integral geometry', Israel J. Math. 10

(1971) 327-338.8. M. GIERTZ, 'A note on a problem of Busemann', Math. Scand. 25 (1969) 145-148.9. E. L. GRINBERG, 'Isoperimetric inequalities for fc-dimensional cross-sections of convex bodies', to

appear.10. D. G. LARMAN and C. A. ROGERS, 'The existence of a centrally symmetric convex body with central

cross-sections that are unexpectedly small', Matematika 22 (1975) 164-175.11. E. LUTWAK, 'Inequalities for Hadwiger's harmonic quermassintegrals', Math. Ann. 280 (1988)

165-175.12. E. LUTWAK, 'Intersection bodies and dual mixed volumes', Adv. in Math. 71 (1988) 232-261.13. C. PETTY, 'Centroid surfaces', Pacific J. Math. 11 (1961) 1535-1547.

Department of Mathematics Computer Science DepartmentTemple University Stanford UniversityPhiladelphia, PA 19122 Stanford, CA 94305USA USA

Current addressWolfram Research Inc.ChampagneIllinois, IL 61826USA

Infinitesimal Aspects of the Busemann-Petty Problem

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