04/10/23 1

DUALITY IN CONVEX GEOMETRY…DUALITY IN CONVEX GEOMETRY…

Richard Gardner

WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY!WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY!

04/10/23 2

Scope of Geometric TomographyScope of Geometric Tomography

Point X-rays

Integral geometry

Local theory of Banach spaces

Robot vision

Stereology and local stereology

Computerized tomographyDiscrete tomography

Parallel X-rays

Minkowski geometry

Imaging

Pattern recognition

Convex geometry

Sections through a fixed point; dual Brunn-Minkowski theory

Projections; classical Brunn-Minkowski theory

?

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ProjectionsProjections and and SectionsSections Theorem 1. (Blaschke and Hessenberg, 1917.) Suppose that

2 ≤ k ≤ n -1, and every k-projection K|S of a compact convex

set in Rn is a k-dimensional ellipsoid. Then K is an ellipsoid.

Theorem 1'. (Busemann, 1955.) Suppose that 2 ≤ k ≤ n -1, and

every k-section K ∩ S of a compact convex set in Rn containing

the origin in its relative interior is a k-dimensional ellipsoid. Then

K is an ellipsoid.

Busemann’s proof is direct, but Theorem 1' can be obtained from

Theorem 1 by using the (not quite trivial) fact that the polar of an

ellipsoid containing the origin in its interior is an ellipsoid.Burton, 1976: Theorem 1' holds for arbitrary compact convex sets

and hence for arbitrary sets.

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The The SupportSupport FunctionFunction

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The The RadialRadial FunctionFunction and and Star BodiesStar Bodies

E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975),

531-538.

Star body: Body star-shaped at o whose radial function is

positive and continuous,

OR one of several other alternative definitions in the literature!

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Polarity – a Projective NotionPolarity – a Projective Notion

Let K be a convex body in Rn with o in int K, and let φ be a nonsingular projective transformation of Rn, permissible for K, such that o is in int φK. Then there is a nonsingular projective transformation ψ, permissible for K*, such that (φK)* = ψK*.

P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, Cambridge University Press, Cambridge, 1971.

).(/1)(* uhu KK

For a convex body K containing the origin in its interior and u in Sn-1,

Then, if S is a subspace, we have

.*)|( * SKSK

We have (φK)* = φ–tK*, for φ in GLn, but…

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Polarity Usually FailsPolarity Usually Fails Theorem 2. (Süss, Nakajima, 1932.) Suppose that 2 ≤ k ≤ n -1 and

that K and L are compact convex sets in Rn. If all k-projections K|S

and L|S of K and L are homothetic (or translates), then K and L are

homothetic (or translates, respectively).

Theorem 2'. (Rogers, 1965.) Suppose that 2 ≤ k ≤ n -1 and

that K and L are compact convex sets in Rn containing the origin in

their relative interiors. If all k-sections K ∩ S and L ∩ S of K and L

are homothetic (or translates), then K and L are homothetic (or

translates, respectively).

Problem 1. (G, Problem 7.1.) Does Theorem 2' hold for star bodies,

or perhaps more generally still?

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WidthWidth and and BrightnessBrightness

width function

)()(

)()(

uhuh

lKVuw

KK

uK

)()( uKVubK

brightness function

uK

u

u

K

K

KSu

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Aleksandrov’s Projection TheoremAleksandrov’s Projection Theorem

For o-symmetric convex bodies K and L,

1

),(|2

1

nS

vKdSv|u(u)bK

. LKu(u)b(u)b LK

Cauchy’s projection formula

Cosine transform of surface area measure of K

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Funk-Minkowski Section TheoremFunk-Minkowski Section Theorem

)()( uKVusK

. LKu(u)s(u)s LK

Funk-Minkowski section theorem:

For o-symmetric star bodies K and L,

section function u

K

uS

nK

n

dvvn 1

1)(1

1

Spherical Radon transform of (n-1)st power of radial function of K

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Projection BodiesProjection Bodies

KK bh

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Shephard’s ProblemShephard’s Problem

• Petty, Schneider, 1967:

For o-symmetric convex bodies K and L,

(i) if L is a projection body and (ii) if and only if n = 2.

)()( LVKVu(u)b(u)b LK

K L

A counterexample for n = 3

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Intersection BodiesIntersection Bodies

KIK sErwin Lutwak

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Busemann-Petty Problem Busemann-Petty Problem (Star Body Version)(Star Body Version)

• Lutwak,1988:

For o-symmetric star bodies K and L,

(i) if K is an intersection body and (ii) if and only if n = 2.

)()( LVKVu(u)s(u)s LK

Hadwiger,1968: A counterexample for n = 3

K L

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Minkowski Minkowski and and RadialRadial Addition Addition

LK ~LK

LL

K K

22 ),(),(2),()( tLLVstLKVsKKVtLsKV 2

~~2

~

),(),(2),()~( tLLVstLKVsKKVtLsKV

LKLK hhh LKLK ~

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Dual Mixed VolumesDual Mixed Volumes

1

21.)()()(

1),...,,( 21

~

nn

S

KKKn duuuun

KKKV

The dual mixed volume of star bodies K1, K2,…, Kn is

E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-

538.

.)(),(

),;,(),(

1)(

),(

inGi

ni dSSKV

incinBiKV

incKV

.)()(1

),;,()(),(

~~

1

inGi

n

S

iKi dSSKVduu

ninBiKVKV

n

Intrinsic volumes and Kubota’s formula:

Dual volumes and the dual Kubota formula (Lutwak, 1979):

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Characterizations of Characterizations of Projection Projection andand Intersection Intersection

Bodies Bodies II Firey, 1965, Lindquist, 1968: A convex body K in Rn is a projection body iff it is a limit (in

the Hausdorff metric) of finite Minkowski sums of n-dimensional o-symmetric ellipsoids.

Definition: A star body is an intersection body if ρL = Rμ for some

finite even Borel measure μ in Sn-1.

Goodey and Weil, 1995: A star body K in Rn is an intersection body iff it is a limit (in the radial metric) of finite radial sums of n-dimensional o-symmetric ellipsoids.

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Characterizations of Characterizations of Projection Projection andand Intersection Intersection

Bodies Bodies IIII Weil, 1976: Let L be an o-symmetric convex body in Rn. An o-

symmetric convex body K in Rn is a projection body iff

for all o-symmetric convex bodies M such that ΠL is contained in ΠM.

),;1,();1,( KnMVKnLV

For a unified treatment, see:

Zhang, 1994: Let L be an o-symmetric star body in Rn. An o-symmetric star body K in Rn is an intersection body iff

),;1,();1,(~~

KnMVKnLV for all o-symmetric star bodies M such that IL is contained in IM.

P. Goodey, E. Lutwak, and W. Weil, Functional analytic characterizations ofclasses of convex bodies, Math. Z. 222 (1996), 363-381.

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Lutwak’s DictionaryLutwak’s Dictionary

Convex bodies Star bodies

Projections Sections through o

Support function hK Radial function ρK

Brightness function bK Section function sK

Projection body ΠK Intersection body IK

Cosine transform Spherical Radon transform

Surface area measure ρKn-1

Mixed volumes Dual mixed volumes

Brunn-Minkowski ineq. Dual B-M inequality

Aleksandrov-Fenchel Dual A-F inequality

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Is Lutwak’s Dictionary Infallible?Is Lutwak’s Dictionary Infallible?

Theorem 3. (Goodey, Schneider, and Weil,1997.) Most convex bodies in Rn are determined, among all convex bodies, up to translation and reflection in o, by their width and brightness functions.

Theorem 3'. (R.J.G., Soranzo, and Volčič, 1999.) The set of star bodies in Rn that are determined, among all star bodies, up to reflection in o, by their k-section functions for all k, is nowhere dense.

Notice that this (very rare) phenomenon does not apply within the class of o-symmetric bodies. There I have no example of this sort.

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Two Two Unnatural Unnatural ProblemsProblems 1. (Busemann, Petty, 1956.) If K and L are o-symmetric convex

bodies in Rn such that sK(u) ≤ sL(u) for all u in Sn-1, is V(K) ≤ V(L)?

2. Problem 2. (R.J.G., 1995; G, Problem 7.6.) If K and L are o-symmetric star bodies in R3 whose sections by every plane through o have equal perimeters, is K =L?

Solved: Yes, n = 2 (trivial), n = 3 (R.J.G., 1994), n = 4 (Zhang,1999).

No, n ≥ 5 (Papadimitrakis, 1992, R.J.G., Zhang, 1994).

Unified: R.J.G., Koldobsky, and Schlumprecht, 1999.

Unsolved.

Theorem 4. (Aleksandrov, 1937.) If K and L are o-symmetric convex bodies in R3 whose projections onto every plane have equal perimeters, then K =L.

SS

K duuSBSKV2

)(2

1),(

~

Yes. R.J.G. and Volčič, 1994.

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Aleksandrov-FenchelAleksandrov-Fenchel InequalityInequality If K1, K2,…, Kn are compact convex sets in Rn, then

.),...,;,(),...,,(1

121

i

jnij

in KKiKVKKKV

Dual Aleksandrov-Fenchel InequalityDual Aleksandrov-Fenchel Inequality If K1, K2,…, Kn are star bodies in Rn, then

,),...,;,(),...,,(1

1

~

21

~

i

jnij

in KKiKVKKKV

with equality when if and only if K1, K2,…, Ki are dilatates of each other.

ni

Proof of dual Aleksandrov-Fenchel inequality follows directly from an extension of Hölder’s inequality.

04/10/23 23

Brunn-MinkowskiBrunn-Minkowski inequalityinequality

If K and L are convex bodies in Rn, then

with equality iff K and L are homothetic.

nnn LVKVLKV /1/1/1 )()()( nnn LVKVLnKV /1/)1( )()();1,(

and(B-M)

(M1)

Dual Brunn-Minkowski Dual Brunn-Minkowski inequalityinequality

If K and L are star bodies in Rn, thennnn LVKVLKV /1/1/1 )()()~(

andnnn LVKVLnKV /1/)1(

~

)()();1,( with equality iff K and L are dilatates.

(d.B-M)

(d.M1)

04/10/23 24

RelationsRelations

.02)~(1);1,( /1~

nLKVLnKV

.1)()( LVKV Suppose that

(B-M)→(M1)

(d.B-M)→(d.M1)

.02)(1);1,( /1 nLKVLnKV H. Groemer, On an inequality of Minkowski for mixed

volumes, Geom. Dedicata. 33 (1990), 117-122.

R.J.G. and S. Vassallo, J. Math. Anal. Appl. 231 (1999), 568-587 and 245 (2000), 502-512.

.1);1,(2)~()12(

1 ~

1

LnKVLKV

n

n nn

.1);1,(2)(1

LnKVLKV

nn

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Busemann Intersection InequalityBusemann Intersection Inequality

with equality if and only if K is an o-symmetric ellipsoid.

If K is a convex body in Rn containing the origin in its interior, then

,)()( 121

n

nn

nn KVIKV

H. Busemann, Volume in terms of concurrent cross-sections,

Pacific J. Math. 3 (1953), 1-12.

1

)(1

)(nS

n duuKVn

IKV

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Generalized BusemannGeneralized BusemannIntersection InequalityIntersection Inequality

with equality when 1 < i < n if and only if K is an o-symmetric ellipsoid and when i = 1 if and only if K is an o-symmetric star body, modulo sets of measure zero.

If K is a bounded Borel set in Rn and 1 ≤ i ≤ n, then

,)()(),(

ini

n

ni

inG

ni KVdSSKV

H. Busemann and E. Straus, 1960; E. Grinberg, 1991; R. E. Pfiefer,

1990; R.J.G., Vedel Jensen, and Volčič, 2003. R.J.G., The dual Brunn-Minkowski theory for bounded Borel sets: Dual

affine quermassintegrals and inequalities, Adv. Math. 216 (2007), 358-386.

04/10/23 27

Petty’s ConjecturedPetty’s ConjecturedProjection InequalityProjection Inequality

with equality if and only if K is an ellipsoid?

Let K be a convex body in Rn. Is it true that

,)()( 121

n

nn

nn KVKV

C. Petty, 1971.

.)()( 1

1

* n

n

n

n KVKV

Petty Projection Inequality:

04/10/23 28

Other RemarksOther Remarks• “Bridges”: Polar projection bodies, centroid bodies, p-

cosine transform, Fourier transform,…

• The Lp-Brunn-Minkowski theory and beyond…

• Gauss measure, p-capacity, etc.

• Valuation theory…

• There may be one all-encompassing duality, but perhaps more likely, several overlapping dualities, each providing part of the picture.