The convex hull of randomly chosen points
Christian Buchta
Salzburg University
Workshop on Stochastic Geometry and its Applications
Rouen, March 30, 2012
Christian Buchta
The convex hull of randomly chosen points
Christian Buchta
The convex hull of randomly chosen points
Christian Buchta
The convex hull of randomly chosen points
Rényi and Sulanke (1963)
2
ENn (K ) = r log n + O(1)
3
Rényi and Sulanke (1964)
2 log n
EAn (K ) = 1 − r
+O
3 n
1
n
Efron (1965)
ENn+1 (K ) = (n + 1) (1 − EAn (K ))
⇒ EAn (K ) = 1 −
Christian Buchta
ENn+1 (K )
n+1
The convex hull of randomly chosen points
B. (1984)
ENn (triangle) = 2
n−1
X
1
k
k=1
" n
#
8 X1
1
1
ENn (square) =
1− k −
3
k
n 2n
2
k=1
Christian Buchta
The convex hull of randomly chosen points
B. (1984)
" n
k !
X1
5
ENn (regular hexagon) = 4
1−
+
k
6
k=1
n n n 1
1
5
1
+2
−1−
+
+
−
6
6
2
n 6n
! n−2
!
#
2n−k − 1 2
n 1 X
n k
k n−2
+2
(−1) 7 + 1
(−1)
2 6n
k
n−k
k=0
Christian Buchta
The convex hull of randomly chosen points
Theorem (B. and Reitzner, 1997):
Let K be a plane convex body of unit area, and consider the
family of all directed chords that divide K into a part of area s
to the left and a part of area 1 − s to the right. The locus of
the midpoints of these chords is a closed curve Ms . Let the
orientation of Ms correspond to the chords rotating in the
counter-clockwise direction. Put
Z
K[s] := 1 −
w (z, Ms )dz,
z∈K \Ms
where w (z, Ms ) is the winding number of Ms about the point
z.
Christian Buchta
The convex hull of randomly chosen points
Then
4
pjk (K ) = jk
3
Z
1
s j−1 (1 − s)k−1 K[s] ds.
0
Especially, the expected number ENn (K ) of vertices of the convex
hull of n points distributed independently and uniformly in K is
given by
Z 1
4
ENn (K ) = n pn−1,1 (K ) = n(n − 1)
s n−2 K[s] ds
3
0
and the expected area EAn (K ) of the convex hull of n such points
by
Z 1
4
s n−1 K[s] ds.
EAn (K ) = 1 − pn,1 (K ) = 1 − n
3 0
Christian Buchta
The convex hull of randomly chosen points
Ms for the regular pentagon
(a)
s =
√
1
− 105
2
≈ 0.276393
1:1
Christian Buchta
0.4
(b) s1=: 0.3
The convex hull of randomly chosen points
Ms for the regular pentagon
0.46
(c) s 1=: 0.2
Christian Buchta
(d)
s =
√
2
+ 255
5
≈ 0.489443
1 : 0.15
The convex hull of randomly chosen points
Ms for the regular pentagon
0.495
(e) s 1=: 0.15
Christian Buchta
0.5
(f) s1 =
: 0.15
The convex hull of randomly chosen points
Theorem (B. and Reitzner, 1997):
Let K be a convex polygon of unit area with vertices
v1 , . . . , vr . Denote by shi the area of the convex hull of the
vertices vh , vh+1 , . . . , vi−1 , vi . Put
(
0
if vh vh+1 and vi vi+1 are parallel,
1
:= shi + sh+1,i+1 − sh,i+1 − sh+1,i
fhi
else.
s s
−s
s
hi h+1,i+1
h,i+1 h+1,i
Christian Buchta
The convex hull of randomly chosen points
Then
r
2r
2X
r
ENn (K ) =
(log n + C ) +
log si,i+2 + −
3
3
3n
i=1
∞
∞
r
2r X B2m 1
2 X n!(m − 1)! X 1
−
−
m +
3
2m n2m 3
(n + m)!
fi,i+2
m=1
m=1
n
i=1
+ O( max (1 − si,i+2 ) ),
i=1,...,r
where the constants B2m are the Bernoulli numbers.
Christian Buchta
The convex hull of randomly chosen points
Classical“ Hypergeometric Function:
”
2 F1 (α1 ,α2 ; β1 ; x)
=1+
=
=
α1 α2 x
α1 (α1 + 1)α2 (α2 + 1) x 2
+
+ ··· =
β1 1!
β1 (β1 + 1)
2!
∞
X
(α1 )k (α2 )k x k
,
(β1 )k
k!
k=0
where (α)k = α(α + 1) · · · (α + k − 1).
Christian Buchta
The convex hull of randomly chosen points
Generalized Hypergeometric Function:
A FB (α1 , α2 , . . . , αA ;β1 , β2 , . . . , βB ; x) =
∞
X
(α1 )k (α2 )k · · · (αA )k
=
k=0
Christian Buchta
xk
.
(β1 )k (β2 )k · · · (βB )k k!
The convex hull of randomly chosen points
Theorem (B. and Reitzner, 2001)
The expected volume V(n) of the convex hull of n random
points chosen independently and uniformly from a tetrahedron
of volume one is given by:
Christian Buchta
The convex hull of randomly chosen points
n−2
n − 2 X
1
1
+
(−1)k
3
(n + 1)
k
(k
+
3)3
k=0
n − 2 4 X
2k2 3 j2 +j3
j1 , . . . , j4 k1 , k2
+...+j =n−2
2
3(n − 1)n
V(n) = 1 −
−
n+1
4
−
9(n − 1)n
2
"
#
j1
5
k1 +k2 +k3 =4
j1 ,...,j5 ,k1 ,k2 ,k3 ≥0
× B(j2 + 2j3 + 3j4 + 3j5 + k2 + 2k3 + 1, 3j1 + 2j2 + j3 + 2k1 + k2 + 1)
× B(n + 1, j5 + k3 + 1) B(2j1 + j2 + k1 + 1, j5 + 2)
× 3 F 2 (j5 + k3 , n + 1, 2j1 + j2 + k1 + 1; j5 + k3 + n + 2, 2j1 + j2 + j5 + k1 + 3; 1)
n − 2 2 2 X
+ 6(n − 1)n
3 j2 +j3
j
,
.
.
.
,
j
l
l
1
4
1
3
j +...+j =n−2
1
5
l1 +l2 =2
l3 +l4 =2
j1 ,...,j5 ,l1 ,l2 ,l3 ,l4 ≥0
× B(j2 + 2j3 + 3j4 + 3j5 + l2 + l4 + 3, 3j1 + 2j2 + j3 + l1 + l3 + 3)
× B(n + 1, j5 + l4 + 1) B(2j1 + j2 + l1 + 1, j5 + 3)
× 3 F2 (j5 + l4 + 1, n + 1, 2j1 + j2 + l1 + 1; j5 + l4 + n + 2, 2j1 + j2 + j5 + l1 + 4; 1).
Christian Buchta
The convex hull of randomly chosen points
The first non-trivial values are:
V(4) =
V(5) =
V(6) =
V(7) =
V(8) =
V(9) =
V(10) =
V(11) =
π2
13
−
720
15015
13
π2
−
288
6006
89π 2
127
−
1680
323323
307
211π 2
−
2880
554268
22829π 2
41369
−
302400
47805615
11129
461π 2
−
67200
817190
641303
3058061π 2
−
332640
4775249765
37723
6445438π 2
−
172800
9116385915
Christian Buchta
≈ 0.0173
≈ 0.0434
≈ 0.0728
≈ 0.1028
≈ 0.1320
≈ 0.1600
≈ 0.1864
≈ 0.2113
The convex hull of randomly chosen points
Identity of the values V(n):
For m = 2, 3, . . .
V(2m + 1) =
m−1
X
k=1
2k
(2
B2k 2m + 1
− 1)
V(2m − 2k + 2),
k
2k − 1
where the constants B2k are the Bernoulli numbers
1
1
1
1
B2 = , B4 = − , B6 = , B8 = − , . . . .
6
30
42
30
Christian Buchta
The convex hull of randomly chosen points
Thus the values are related by:
5
V(4),
2
7
35
V(7) = V(6) −
V(4),
2
4
9
V(9) = V(8) − 21 V(6) + 63 V(4),
2
11
165
2805
V(11) =
V(10) −
V(8) + 231 V(6) −
V(4).
2
4
4
V(5) =
Christian Buchta
The convex hull of randomly chosen points
Bárány and B. (1993)
ENn (P) =
F (P)
d−1
d−2
log
n+O
log
n
log
log
n
(d + 1)d−1 (d − 1)!
Christian Buchta
The convex hull of randomly chosen points
Groeneboom (1988)
Nn − 23 r log n D
q
→ N (0, 1)
10
r
log
n
27
Cabo and Groeneboom (1994)
Dn − 23 rn−1 log n D
q
→ N (0, 1)
100 −1
rn
log
n
189
Christian Buchta
The convex hull of randomly chosen points
Efron (1965)
ENn+1
EVn
=1−
vol K
n+1
B. (2000/2005)
k Y
Nn+k
=E
1−
n+i
(vol K )k
i=1
EVnk
Thus the k th moment of Vn can be expressed by the first k
moments of Nn+k :
EVnk
1
1
= 1−
+ ··· +
ENn+k
n+1
n+k
(vol K )k
1
EN k .
+ · · · + (−1)k
(n + 1) · · · (n + k) n+k
Christian Buchta
The convex hull of randomly chosen points
Christian Buchta
The convex hull of randomly chosen points
Christian Buchta
The convex hull of randomly chosen points
Bárány, Rote, Steiger and Zhang (2000)
(n)
qn =
2n
n!(n + 1)!
B. (2006)
(n)
qk
1
i1 (i1 + i2 ) · · · (i1 + i2 + . . . + ik )
i1 i2 · · · ik
×
(i1 + 1)(i1 + i2 + 1) · · · (i1 + i2 + . . . + ik + 1)
= 2k
X
where i1 , i2 , . . . , ik ∈ N such that i1 + i2 + . . . + ik = n
Christian Buchta
The convex hull of randomly chosen points
Recurrence formulae:
2
(n−1)
(n)
q
qn =
n(n + 1) n−1
n−1
X
2
(n)
(j)
qk =
(n − j)qk−1
n(n + 1)
j=k−1
n(n + 1) (n)
(n − 2)(n − 1) (n−2)
(n−1)
(n−1)
qk −(n −1)nqk
+
qk
= qk−1
2
2
Christian Buchta
The convex hull of randomly chosen points
Hence,
(n − 2)(n − 1)
n(n + 1)
m
m
ENnm − (n2 − n + 1)ENn−1
+
ENn−2
2
2
m X
m
m−j
=
ENn−1
.
j
j=1
We study now the difference equation
n(n + 1)
(n − 2)(n − 1)
f (n)−(n2 −n+1)f (n−1)+
f (n−2) = g (n).
2
2
Christian Buchta
The convex hull of randomly chosen points
Expected value and variance:
n
2X1 1
ENn =
+
3
k
3
k=1
ENn2
4
=
9
n
X
1
k
!2
k=1
var Nn =
10
27
n
n
22 X 1 4 X 1
4
25
+
+
+
−
27
k
9
k 2 27 9(n + 1)
k=1
n
X
k=1
1 4
+
k
9
Christian Buchta
k=1
n
X
k=1
1
4
28
+
−
k 2 27 9(n + 1)
The convex hull of randomly chosen points
Third Moment:
ENn3
=
8
27
−
−
16
9
8
9n
n
X
1
k
k=1
n
X
k2 =1
n
X
k=1
!3
32
+
27
n
X
1
k
!2
k=1
n
+
8X 1
106
−
2
9
k
81
!
k=1
k2
n
n
1 X
1
32 X 1
16 X 1
91
+
+
+
2
2
3
k1 27
k
27
k
81
k2
k1 =1
k=1
k=1
1
16
−
k
27(n + 1)
Asymptotic behaviour of the mth moment:
m
2
m
ENn =
log n
+ O logm−1 n
3
Christian Buchta
The convex hull of randomly chosen points
n
X
1
k
k=1
Christian Buchta
The convex hull of randomly chosen points
Position of the points: Type (a)
(n)
pa
=
(n − 2)(n − 3)
n(n − 1)
Christian Buchta
The convex hull of randomly chosen points
Position of the points: Type (b)
(n)
pb =
Christian Buchta
4(n − 2)
n(n − 1)
The convex hull of randomly chosen points
Position of the points: Type (c)
(n)
pc
=
Christian Buchta
2
n(n − 1)
The convex hull of randomly chosen points
Christian Buchta
The convex hull of randomly chosen points
A3
A2
A0
A4
A0 = 1 −
A1 =
A2 =
A3 =
A4 =
1
2
1
2
1
2
1
2
1
2
A1
(x1 + x2 + x3 + x4 − x1 x2 − x2 x3 − x3 x4 − x4 x1 )
x1 (1 − x2 )
x2 (1 − x3 )
x3 (1 − x4 )
x4 (1 − x1 )
Christian Buchta
The convex hull of randomly chosen points
A3
A2
A0
A4
A1
4
(n)
Pk1 ,k2 ,k3 ,k4 (x1 , x2 , x3 , x4 ) = (n − 4)!
X
r0 +···+r4 =n−4
r0 ≥0
r1 ≥k1 ,...,r4 ≥k4
Christian Buchta
1 r0 Y 1 (ri ) ri
A
q A
r0 ! 0
ri ! ki i
i=1
The convex hull of randomly chosen points
Probability that the convex hull in the square has exactly k
vertices:
(n)
(n) (n)
(n) (n)
(n) (n)
pk (square) = pa pk|a + pb pk|b + pc pk|c
Christian Buchta
The convex hull of randomly chosen points
Probability that the convex hull in the square is of a certain type
and has exactly k vertices (n = 7):
k
3
4
5
6
7
Σ
a
0
65
1008
155
672
475
3024
145
6048
10
21
b
137
10080
1811
12600
1046
4725
223
2520
1361
151200
10
21
c
1
1008
437
25200
1621
75600
109
15120
1
1575
1
21
Σ
7
480
203
900
3409
7200
91
360
121
3600
1
Christian Buchta
The convex hull of randomly chosen points
Expected value and variance of Nn (triangle):
ENn (triangle) = 2
n−1
X
1
k
k=1
var Nn (triangle) =
Christian Buchta
n−1
n−1
k=1
k=1
10 X 1 4 X 1
−
9
k
3
k2
The convex hull of randomly chosen points
Recent important contributions are due to:
Imre Bárány
Pierre Calka
Xavier Goaoc
Matthias Reitzner
Tomasz Schreiber
...
Christian Buchta
The convex hull of randomly chosen points