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DUALITY IN CONVEX GEOMETRY…
WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY!
Richard Gardner
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Scope of Geometric Tomography
Computerized tomography
Discrete tomography
Convex
geometry
Integral
geometry
Point
X-rays
Parallel
X-rays
Projections;
classical BrunnMinkowski theory
Minkowski
geometry
Local theory of Banach spaces
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Robot vision
Imaging
Sections through a
fixed point; dual
Brunn-Minkowski
Pattern
theory
recognition
Stereology and local
stereology
2
Projections and Sections
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 Support Function
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The Radial Function and Star Bodies
Star body: Body star-shaped at o whose radial function is
positive and continuous,
OR one of several other alternative definitions in the literature!
E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538.
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Polarity – a Projective Notion
For a convex body K
containing the origin in its
interior and u in Sn-1,
 K * (u )  1 / hK (u ).
Then, if S is a subspace,
we have
K *  S  (K | S ) *.
We have (φK)* = φ–tK*, for φ in GLn, but…
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*.
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P. McMullen and G. C. Shephard, Convex polytopes and the upper bound
conjecture, Cambridge University Press, Cambridge, 1971.
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Polarity 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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Width and Brightness
Su K
K
width function
wK (u )  V ( K l u )
 hK (u )  hK (u )
u
K
brightness function
bK (u )  V ( K u  )
u
K u
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Aleksandrov’s Projection Theorem
For o-symmetric convex bodies K and L,
bK (u)  bL(u) u  K  L.
1
bK (u)  |u  v | dS ( K , v)
2 S n 1
Cauchy’s projection formula
Cosine transform of surface area measure of K
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Funk-Minkowski Section Theorem
u
K

section function

sK ( u )  V ( K  u )
1
n 1

 K (v) dv

n  1 S n 1 u 
Spherical Radon transform of (n-1)st power of radial function of K
Funk-Minkowski section theorem:
For o-symmetric star bodies K and L,
s K (u)  s L(u) u  K  L.
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Projection Bodies
hK  bK
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Shephard’s Problem
• Petty, Schneider, 1967:
For o-symmetric convex bodies K and L,
bK (u)  bL(u) u  V ( K )  V ( L)
(i) if L is a projection body and (ii) if and only if n = 2.
K
L
A counterexample for n = 3
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Intersection Bodies
Erwin Lutwak
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 IK  s K
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Busemann-Petty Problem
(Star Body Version)
• Lutwak,1988:
For o-symmetric star bodies K and L,
sK (u)  sL(u) u  V ( K )  V ( L)
(i) if K is an intersection body and (ii) if and only if n = 2.
K
L
Hadwiger,1968: A counterexample for n = 3
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Minkowski and Radial Addition
L
L
K
KL
K
K~
L
 K ~ L   K   L
hK  L  hK  hL
V (sK  tL)  V ( K , K )s  2V ( K , L)st  V ( L, L)t
~
~
~
V ( sK ~
 tL)  V ( K , K ) s 2  2V ( K , L) st  V ( L, L)t 2
2
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Dual Mixed Volumes
The dual mixed volume of star bodies K1, K2,…, Kn is
1
V ( K1 , K 2 ,..., K n )    K1 (u )  K 2 (u )   K n (u )du.
n S n1
~
E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538.
Intrinsic volumes and Kubota’s formula:
n
1
Vi ( K ) 
V ( K , i; B , n  i ) 
V ( K S )dS .

c(n, i )
 i c(n, i ) G ( n,i )
Dual volumes and the dual Kubota formula (Lutwak, 1979):
n
1
i
V i ( K )  V ( K , i; B, n  i )    K (u ) du 
n S n1
i
~
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 V ( K  S )dS.
G ( n ,i )
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Characterizations of Projection
and Intersection Bodies I
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 Projection
and Intersection Bodies II
Weil, 1976: Let L be an o-symmetric convex body in Rn. An osymmetric convex body K in Rn is a projection body iff
V ( L, n  1; K )  V ( M , n  1; K ),
for all o-symmetric convex bodies M such that ΠL is contained in ΠM.
Zhang, 1994: Let L be an o-symmetric star body in Rn. An osymmetric star body K in Rn is an intersection body iff
~
~
V ( L, n  1; K )  V ( M , n  1; K ),
for all o-symmetric star bodies M such that IL is contained in IM.
For a unified treatment, see:
P. Goodey, E. Lutwak, and W. Weil, Functional analytic characterizations of
classes of convex bodies, Math. Z. 222 (1996), 363-381.
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Lutwak’s Dictionary
Convex bodies
Projections
Star bodies
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?
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 Unnatural Problems
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)?
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.
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.
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
Yes. R.J.G. and Volčič, 1994.
through o have equal perimeters, is K =L?
~
1
V (K  S , B  S ) 
 K (u )du
Unsolved.

2 S 2 S
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Aleksandrov-Fenchel Inequality
If K1, K2,…, Kn are compact convex sets in Rn, then
i
V ( K1 , K 2 ,..., K n )i  V ( K j , i; K i 1 ,..., K n ).
j 1
Dual Aleksandrov-Fenchel Inequality
If K1, K2,…, Kn are star bodies in Rn, then
~
i
~
V ( K1 , K 2 ,..., K n )i  V ( K j , i; K i 1 ,..., K n ),
j 1
with equality when i  n if and only if K1, K2,…, Ki are
dilatates of each other.
Proof of dual Aleksandrov-Fenchel inequality follows directly
from an extension of Hölder’s inequality.
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Brunn-Minkowski inequality
If K and L are convex bodies in Rn, then
V ( K  L)
1/ n
and
 V (K )
1/ n
V ( K , n  1; L)  V ( K )
 V ( L)
( n1) / n
1/ n
1/ n
V ( L)
(B-M)
(M1)
with equality iff K and L are homothetic.
Dual Brunn-Minkowski inequality
If K and L are star bodies in Rn, then
V ( K ~ L)1/ n  V ( K )1/ n  V ( L)1/ n
and
~
V ( K , n  1; L)  V ( K )
( n 1) / n
(d.B-M)
1/ n
V ( L)
(d.M1)
with equality iff K and L are dilatates.
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Relations
Suppose that
V ( K )  V ( L)  1.
V ( K , n  1; L)  1  V ( K  L)
1/ n
 2  0.
H. Groemer, On an inequality of Minkowski for mixed
volumes, Geom. Dedicata. 33 (1990), 117-122.
(B-M)→(M1)
~
~
1/ n
V ( K , n  1; L)  1  V ( K  L)  2  0.
(d.B-M)→(d.M1)
R.J.G. and S. Vassallo, J. Math. Anal. Appl.
231 (1999), 568-587 and 245 (2000), 502-512.




1
n
  V ( K  L)  2  V ( K , n  1; L)  1.
n
~
 n 1 
~
n


V
(
K

L
)

2
 V ( K , n  1; L)  1.
n 1

 n(2  1) 
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Busemann Intersection Inequality
If K is a convex body in Rn containing the origin in its
interior, then
 nn1
V ( IK )  n2 V ( K ) n1 ,
n
with equality if and only if K is an o-symmetric ellipsoid.
H. Busemann, Volume in terms of concurrent cross-sections,
Pacific J. Math. 3 (1953), 1-12.
1
V ( IK )   V ( K  u  ) n du
n S n 1
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Generalized Busemann
Intersection Inequality
If K is a bounded Borel set in Rn and 1 ≤ i ≤ n, then

i
Vi ( K  S ) dS 
Vn ( K ) ,


G ( n ,i )
n
n
i
i
n
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.
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.
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Petty’s Conjectured
Projection Inequality
C. Petty, 1971.
Let K be a convex body in Rn. Is it true that

V ( K ) 

n
n 1
n2
n
V (K )
n 1
,
with equality if and only if K is an ellipsoid?
Petty Projection Inequality:
n
 n 
 V ( K )1n .
V ( K )  
  n1 
*
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Other Remarks
•
“Bridges”: Polar projection bodies, centroid bodies, pcosine 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.
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