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)

8

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).

10

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.

12

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 .

13

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).

14

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.

15

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, ·).

18

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, ·).

19

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, ·).

20