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Determination of Convex Bodiesby
Projection Functions
Wolfgang Weil
U Karlsruhe
weil@math.uni-karlsruhe.de
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 ∈ Lnj .
↑i-th intrinsic volume oforthogonal 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, ·).
1
The proof is based on the following steps:
• The Cauchy-Kubota formula shows that πij(K, ·) determines
πi n−1(K, ·).
• The integral representation
πi n−1(K, u⊥) =1
2
∫Sn−1
|〈u, x〉|Si(K; dx), u ∈ Sn−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; ·) deter-
mines 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 as-
sumed?
(2) Are there variants of πij(K, ·), which allow the determination
of general (non-symmetric) bodies?
3
Problem by Hermann Dinges for my diploma thesis (1968, Frank-
furt):
Let Z be a body with support function
h(Z, u) =∫Sn−1
|〈u, x〉|ρ(Z; dx), u ∈ Sn−1.
The behavior of h(Z, ·) on a small ‘cap’ Uε(u) ⊂ Sn−1 (with
centre u) should be determined by the behavior of ρ(Z; ·) in the
orthogonal ‘zone’
Uε(u⊥) = v ∈ Sn−1 : v⊥w for some w ∈ Uε(u).
If we let the Uε(u) vary over a whole zone Uε(v⊥), u ∈ v⊥, v
fixed, the corresponding zones Uε(u⊥) cover Sn−1. Show that
therefore the knowledge of h(Z, ·) in Uε(v⊥) determines ρ(Z; ·)uniquely (and so also Z).
4
R. Schneider, Zu einem Problem von Shephard uber die Projek-
tionen konvexer Korper, Math. Z. 101, 71–82 (1967):
The Cosine Transform C is a continuous bijection on C∞e (Sn−1).
(⇒ Answer to above problem is no!)
5
Theorem (Schneider/W. 1970). Let K, M be convex bodies
in Rn, n odd, which have a vertex with normal u ∈ Sn−1 and are
centrally symmetric to 0. If, for some ε > 0,
Vn−1(K|v⊥) = Vn−1(M |v⊥), for all v ∈ Uε(u⊥),
then K = M .
The theorem is wrong in even dimensions or if only one of the
bodies has a vertex.
6
7
Theorem (Schneider/W. 1970). Let K, M be convex bodies
in Rn, n odd, which have a vertex with normal u ∈ Sn−1 and are
centrally symmetric to 0. If, for some ε > 0,
Vn−1(K|v⊥) = Vn−1(M |v⊥), for all v ∈ Uε(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, ·).
9
Flag representations of convex bodies
Motivation:
For generalized zonoids K, M , and j = 1, ..., n− 1, we have
πjj(K, ·) = cnj
∫Ln
j
|〈L, ·〉|ρj(K; dL)
and
V (K[j], M [n− j]) = dnj
∫Ln
n−j
πjj(K, N⊥)ρn−j(M ; dN)
= enj
∫Ln
j
∫Ln
n−j
|〈L, N⊥〉|ρn−j(M ; dN)ρj(K; dL).
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For arbitrary (non-symmetric) convex bodies K, M a correspond-
ing formula is classical for j = 1:
V (K[1], M [n− 1]) =1
n
∫Sn−1
h(K, u)Sn−1(M ; du).
What about j > 1 ?
11
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
Fnk := (u, L) ∈ Sn−1 × Ln
k : u⊥L, 1 ≤ k ≤ n− 1,
let
Ωk(K;A) :=∫Ln
k+1
S′k(K|N, A|N)dN,
where A ⊂ Fnn−1−k runs through the Borel sets and
A|N := u ∈ Sn−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.)
There exists a measurable function fk on Fnn−1−k × Fn
k−1 such
that
V (K[k], M [n− k])
=∫Fn
n−1−k
∫Fn
k−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 ∈ Lnj , one gets
Vj(K|L) =∫Fn
n−1−j
gj(u, N ;L)Ωj(K; d(u, N)),
with some function gj on Fnn−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;Sn−1), 1 ≤ i < n.
We therefore define a directed projection function vij(K;L, u),for 1 ≤ i < j ≤ n− 1, L ∈ Ln
j and almost all u ∈ L ∩ Sn−1, by
vij(K;L, u) := cijS′i
(K|L;u+ ∩ L
).
Here, u+ := v ∈ Sn−1 : 〈u, v〉 ≥ 0.(Groemer (1997) discussed the function v12(K, ·), for n = 3.)vij(K; ·) can be interpreted as a function on Fn
j−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.
16
(2) Second approach (Goodey/W. 2005):
We have
Vi(K) = cin
∫Nor(K)
〈x, u〉〈u, v〉Θi−1(K; d(x, v))
+ din
∫Sn−1
〈u, v〉2Si(K; dv) , 1 ≤ i ≤ n.
(Here, u ∈ Sn−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 projectionfunction
pij(K;L, u) = cij
∫Nor(K|L)
1u+(v)〈x, u〉〈u, v〉Θ′i−1(K|L; d(x, v))
+ dij
∫u+∩L
〈u, v〉2S′i(K|L; dv) ,
(for 1 ≤ i ≤ j ≤ n− 1, L ∈ Lnj and u ∈ L ∩ Sn−1).
17
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 S′1(K|L; ·), whereas p1j(K;L, ·) is the
hemispherical transform of the support function h(K|L, ·).
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Projection averages
Let vj(K, u) and pj(K, u) be the averages of v1j(K;L, u) resp.
p1j(K;L, u) over all L ∈ Lnj which contain u (we only have results
for this linear case i = 1).
Theorem. Let K, M ⊂ Rn be convex bodies and 2 ≤ j < 2n−35 or
n−22 ≤ j ≤ n− 1. If
vj(K, ·) = vj(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
v2i+1(K, ·) = v2i+1(M, ·).
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Theorem. Let K, M ⊂ Rn be convex bodies and 1 ≤ j ≤ n2 +1 or
2n+13 < j ≤ n− 1, for n 6= 4 (resp. j = 2 if n = 4). If
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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