Determination of Convex Bodies
by
Projection Functions
Wolfgang Weil
U Karlsruhe
[email protected]
Florence, May 16–20, 2005
The classical uniqueness result
For 1 ≤ i ≤ j ≤ n − 1, the (i, j)-th projection function πij (K, ·)
of a convex body K is defined by
πij (K, ·) := Vi(K|L),
L ∈ Ln
j.
↑
i-th intrinsic volume of
orthogonal projection
A classical result of Alexandrov (1937) yields that a centrally
symmetric body K (of dimension ≥ i + 1) is uniquely determined
(up to translation) by the projection function πij (K, ·).
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The proof is based on the following steps:
• The Cauchy-Kubota formula shows that πij (K, ·) determines
πi n−1(K, ·).
• The integral representation
1
πi n−1(K, u⊥) =
|hu, xi|Si(K; dx),
n−1
2 S
Z
u ∈ S n−1,
and the injectivity properties of the Cosine Transform show
that the surface area measure Si(K; ·) of K is determined (here
the symmetry of K comes in).
• For dim(K) ≥ i + 1, the surface area measure Si(K; ·) determines K (up to translation).
2
Natural questions
(1) What do we know about K, if only partial knowledge of
πij (K, ·) or knowledge of modified functions (averages) is assumed?
(2) Are there variants of πij (K, ·), which allow the determination
of general (non-symmetric) bodies?
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Problem by Hermann Dinges for my diploma thesis (1968, Frankfurt):
Let Z be a body with support function
h(Z, u) =
Z
S n−1
|hu, xi|ρ(Z; dx),
u ∈ S n−1.
The behavior of h(Z, ·) on a small ‘cap’ U(u) ⊂ S n−1 (with
centre u) should be determined by the behavior of ρ(Z; ·) in the
orthogonal ‘zone’
U(u⊥) = {v ∈ S n−1 : v⊥w for some w ∈ U(u)}.
If we let the U(u) vary over a whole zone U(v ⊥), u ∈ v ⊥, v
fixed, the corresponding zones U(u⊥) cover S n−1. Show that
therefore the knowledge of h(Z, ·) in U(v ⊥) determines ρ(Z; ·)
uniquely (and so also Z).
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R. Schneider, Zu einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101, 71–82 (1967):
The Cosine Transform C is a continuous bijection on Ce∞(S n−1).
(⇒ Answer to above problem is no!)
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Theorem (Schneider/W. 1970). Let K, M be convex bodies
in Rn, n odd, which have a vertex with normal u ∈ S n−1 and are
centrally symmetric to 0. If, for some > 0,
Vn−1(K|v ⊥) = Vn−1(M |v ⊥),
for all v ∈ U(u⊥),
then K = M .
The theorem is wrong in even dimensions or if only one of the
bodies has a vertex.
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7
Theorem (Schneider/W. 1970). Let K, M be convex bodies
in Rn, n odd, which have a vertex with normal u ∈ S n−1 and are
centrally symmetric to 0. If, for some > 0,
Vn−1(K|v ⊥) = Vn−1(M |v ⊥),
for all v ∈ U(u⊥),
then K = M .
Schneider/W. 1983: Survey on zonoids (green book)
Schneider/W. 1986: Kinematic formula for curvature measures
(basic paper for translative integral geometry)
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Theorem (Goodey/Schneider/W. 1995). For 1 < i < n − 1,
there are convex bodies K in Rn, n ≥ 3, which are not centrally
symmetric and which are uniquely determined (up to translation
and reflection) by πii(K, ·).
Moreover, there are also centrally symmetric bodies M which are
uniquely determined (up to translations) within all convex bodies
by πii(K, ·).
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Flag representations of convex bodies
Motivation:
For generalized zonoids K, M , and j = 1, ..., n − 1, we have
πjj (K, ·) = cnj
Z
Ln
j
|hL, ·i|ρj (K; dL)
and
V (K[j], M [n − j]) = dnj
= enj
Z
Ln
Z n−j
Z
πjj (K, N ⊥)ρn−j (M ; dN )
n
Ln
j Ln−j
|hL, N ⊥i|ρn−j (M ; dN )ρj (K; dL).
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For arbitrary (non-symmetric) convex bodies K, M a corresponding formula is classical for j = 1:
1
V (K[1], M [n − 1]) =
h(K, u)Sn−1(M ; du).
n−1
n S
Z
What about j > 1 ?
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Flag measures
Schneider (1978): Integral geom. interpr. of curvature meas.
→ W. (1981): Extended curvature measures on flats
→ Kropp (1990): Extended area measures (flag measures)
(→ common generalization: Extended support measures)
For
Fkn := {(u, L) ∈ S n−1 × Ln
k : u⊥L},
1 ≤ k ≤ n − 1,
let
Ωk (K; A) :=
Z
Ln
k+1
Sk0 (K|N, A|N )dN,
n
where A ⊂ Fn−1−k
runs through the Borel sets and
A|N := {u ∈ S n−1 : (u, L) ∈ A for some L⊥N }.
⇒ Sk (K; ·) is the image of Ωk (K; ·) under (u, L) 7→ u.
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Representation of mixed volumes
(Project by Goodey, Hinderer, Hug, Rataj, W.)
n
n
There exists a measurable function fk on Fn−1−k
× Fk−1
such
that
V (K[k], M [n − k])
=
Z
n
Fn−1−k
Z
n
Fk−1
fk (u, L; v, N )Ωn−k (M ; d(v, N ))Ωk (K; d(u, L)),
for all convex bodies K, M .
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Representation of projection functions
For M = unit ball in L⊥, L ∈ Ln
j , one gets
Vj (K|L) =
Z
n
Fn−1−j
gj (u, N ; L)Ωj (K; d(u, N )),
n
with some function gj on Fn−1−j
× Ln
j (Hinderer, 2002).
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Directed projection functions (flag functions)
(1) First approach (Goodey/W. 2004):
We have
Vi(K) = cinSi(K; S n−1),
1 ≤ i < n.
We therefore define a directed projection function vij (K; L, u),
n−1 , by
for 1 ≤ i < j ≤ n − 1, L ∈ Ln
j and almost all u ∈ L ∩ S
0
+
vij (K; L, u) := cij Si K|L; u ∩ L .
Here, u+ := {v ∈ S n−1 : hu, vi ≥ 0}.
(Groemer (1997) discussed the function v12(K, ·), for n = 3.)
n .
vij (K; ·) can be interpreted as a function on Fj−1
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Theorem. If 1 ≤ i < j ≤ n − 1 and K, M are convex bodies of
dimension at least i + 1 with
vij (K; ·) = vij (M ; ·),
then K and M are translates.
There is also a corresponding stability result.
Disadvantages: (a) Determination only up to translation.
(b) Approach does not work for i = j.
(c) K may be ‘over-determined’ by such flag functions.
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(2) Second approach (Goodey/W. 2005):
We have
Vi(K) = cin
Z
Nor(K)
hx, uihu, viΘi−1(K; d(x, v))
+ din
Z
S n−1
hu, vi2Si(K; dv) ,
1 ≤ i ≤ n.
(Here, u ∈ S n−1 is arbitrary and Θj (K; ·) is the j-th support measure on the
normal bundle Nor(K) of K.)
We therefore define a (second kind of) directed projection
function
pij (K; L, u) = cij
Z
Nor(K|L)
Z
+ dij
1u+ (v)hx, uihu, viΘ0i−1(K|L; d(x, v))
hu, vi2Si0 (K|L; dv) ,
u+ ∩L
(for 1 ≤ i ≤ j ≤ n − 1, L ∈ Lnj and u ∈ L ∩ S n−1 ).
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Theorem. If 1 ≤ i ≤ j ≤ n − 1 and K, M are convex bodies of
dimension at least i + 1 with
pij (K; ·) = pij (M ; ·),
then K = M .
There is no corresponding stability result yet.
Again, K may be over-determined by pij (K; ·).
For i = 1, v1j (K; L, ·) is the hemispherical transform of the
first surface area measure S10 (K|L; ·), whereas p1j (K; L, ·) is the
hemispherical transform of the support function h(K|L, ·).
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Projection averages
Let v j (K, u) and pj (K, u) be the averages of v1j (K; L, u) resp.
p1j (K; L, u) over all L ∈ Ln
j which contain u (we only have results
for this linear case i = 1).
or
Theorem. Let K, M ⊂ Rn be convex bodies and 2 ≤ j < 2n−3
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n−2 ≤ j ≤ n − 1. If
2
v j (K, ·) = v j (M, ·),
then K, M are translates.
On the other hand, for i = 1, 2, . . . , there are convex bodies K, M
in R5i+4, which are not translates, with
v 2i+1(K, ·) = v 2i+1(M, ·).
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Theorem. Let K, M ⊂ Rn be convex bodies and 1 ≤ j ≤ n
2 + 1 or
2n+1
< j ≤ n − 1, for n 6= 4 (resp. j = 2 if n = 4). If
3
pj (K, ·) = pj (M, ·),
then K = M .
On the other hand, for i = 1, 2, . . . , there are convex bodies
K 6= M in R3i+1 with
p2i+1(K, ·) = p2i+1(M, ·).
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