Rend. Sem. Mat. Univ. Poi. Torino
Voi. 49, 3 (1991)
F. *Affent ranger
THE CONVEX HULL OF R A N D O M POINTS WITH
SPHERICALLY SYMMETRIC DISTRIBUTIONS*
A b s t r a c t . Let X\, . . ., Xn be n independent, identically distributed random
points in (/-dimensionai Euclidean space, Hn their convex hull and En(ip)
the
mathematical expectation of ìj) o Hn for some measurable function defined on
convex polytopes. In the present paper, we consider the case in which ìj) is the
i th quermassintegral of Minkowski (ì: = 0 , 1 , . .. , d — 1) and À"i,. . . , Xn are
selected according to certain spherically symmetric distributions.
Precise asymptotic formulae of En{ij)) for n tending to infìnity, are
established.
1. Introduction
There is an increasing literature on convex hulls of independent random
points, starting with the well-known papers of R.ényi Sz Sulanke [18], [19] on
the piane case, which have stimulated various generalizations and extensions
to higher dimensions. A survey up to 1987 is given in Schneider [22]; more
recent contributions are due to Affentranger and Wieacker [2], Bàràny [3],
Dwyer [8], [9], and others.
Let X\,... ,Xn be n independent, identically distributed (i.i.d.) random
points in d-dimensionai Euclidean space Md(d > 2), Hn their convex hull,
and En(ip) the mathematical expectation of ifi o Hn for some (measurable)
function tp defined on convex polytopes. Typical functions tp studied in
the literature are numbers of vertices and facets, volume, surface area,
and mean width. Probability distributions for the random points that
Research supported by the Swiss National Foundation
360
have been considered include uniform distributions in convex bodies or on
their boundaries, particulary for balls, normal distributions, and spherically
symmetric distributions. Explicit formulae for En(ip) are rare but, depending
on the function and the probability distribution considered, one has some
information on the order of En(ip) for n tending to infinity, and in some cases
even precise asymptotic formulae.
In the present paper, the i.i.d. random points will be selected according
to certain spherically symmetric distributions. The functional ip considered
will be the ith quermassintegral or Minkowski functional W^ , i = 0 , 1 , . . . , d.
This includes and generalizes the cases of volume, surface area, and mean
width, which are Constant multiples of WQ, W^ ' and W^JV respectively.
For information on quermassintegrals in the geometry of convex bodies,
we refer to Bonnesen-Fenchel [5], Hadwiger [12], Leichtweifi[13], and for their
role in integrai geometry and stochastic geometry, to Santaló [21], Matheron
[14], Stoyan-Kendall-Mecke [24].
We are mainly interested in exact asymptotic formulae for En{W^
when the number n of random points tends to infinity.
')
A typical result is the following assertion (see Corollary 2).
The expected value of the ith quermassintegral W± of the convex hull
of n i.i.d. random points, each selected according to the uniform distribution
in the e?-dimensionai unit ball B satisfìes
ffjVi-vlVc»^,
i = i,...,d-i,
as n tends to infinity. Here e is a Constant (explicitly given below) which
depends only on i and d.
In treating the quermassintegrals, we use integral-geometric projection
formulae. Under projection, uniform random points in a ball are transformed
into beta-distributed points in a lower dimensionai ball. Therefore, we
consider also convex hulls of random points in balls selected according to a
beta-type distribution. In general, our methods lead to conclusive results for
such distributions for which a certain integral-geometric transformation, which
has to be applied to the joint distribution of d i.i.d. random points, yields
a product measure of a special forni. Ruben and Miles [20], where a precise
formulation can be found, have proved that the only spherically symmetric
and absolutely continuous probability distributions with this property are
-
361
the multivariate normal and certain Cauchy-type and beta-type distributions.
Hence, it seems naturai to study these distributions in our context.
We note here that, using the isomorphism between convex subsets of
Euclidean space IR and certain continuous functions on the unit sphere
S
, Eddy and Gale [1*0] have found an interesting connection between
the distribution of the maximum of a collection of one-dimensional random
variables and the distribution of the convex hull of multidimensional random
variables, which leads to the same three distributions treated by Ruben and
Miles [20] (see also the contributions of Carnai [7] and Dwyer [8]).
We have organized this article as follows.
Section 3 deals with random points in the unit ball, each selected
according to a beta-type distribution.
Theorem 1 gives the asymtotic
expansion of a class of useful fimctionals introduced by Wieacker [25]. We have
chosen this approach in order to avoid a separate study of similar particular
cases. The asymptotic behaviour of the ith quermassintegral W^
of the
convex hull of n i.i.d. beta-distributed random points is given in Theorem 2.
Section 4 deals with normally distributed random points in M.d and is
organized in analogy to Section 3.
For the sake of completeness, and since the results follow readily from
those of Section 3, we consider random points uniformly distributed on the
(d — l)-dimensional unit sphere; this is done in Section 5.
The somewhat compiicated explicit asymptotic Constants of Theorem 1
and Corollary 1 are listed in Appendix I.
Our methods easily extend to the case of Cauchy-distributed random
points, but no new results are to be expected from the geometrie viewpoint;
therefore we have restricted our attention to the other cases.
We start with some preliminary remarks.
2. S o m e Preliminaries
We first list some notations which will be used in the following.
R
d-dimensional Euclidean space with || • || and origin 0 (d > 2),
A^
Lebesgue measure in E ,
Bd
d-dimensional unit ball centred at 0,
362
Sd~x
(d— 1)-dimensionai unit sphere centred at 0,
T(x)
Gamma function,
B(x,y)
Beta function,
Kd
= Trd/2/Y(
u>d-i
— ditd : (d — 1)-dimensionai content of 5
): d-dimensionai
volume of Bd,
1
.
Let K be a convex body (compact convex set with nonempty interior)
d
in R . The Minkowski quermassintegrals WQ (K),...
defined as the coefficients in the Steiner polynomial
i=0 ^
where V(K£)
,W^
(K) of K can be
^
is the volume of the parallel body Ke of K at the distance e.
Finally, we write f(x) ~ g(x) to mean f(x) = (1 + o(l))g(x) for a; tending
to infinity. Ali asymptotic expressions appearing in the text refer to n -> oo.
3. R a n d o m points in t h e unit ball
Let Hn be the convex hull of n i.i.d. random points X i , . . . , X w , each
selected according to a multivariate beta-type distribution.
For convenience, we choose the distribution such that for each Borei subset
A of Bd we ha ve
(3.1) Prob(X G A) =
/ (1 - | | x | | 2 ) « f a d ( x ) ,
\
q > -1 .
Let F(Hn) be the set of facets of Hn and denote by 7]jr the distance
between the affine huir aff F of F and the tangent hyperplane of B parallel
to aff F and separated from Hn by aff F. Then, for s E l + and / G N U {0},
define
FeF(Hn)
363
This class of functionals was introduced by Wieacker [25]. for instance,
TQ o and TQ ^ are the number of facets and the surface area of the convex hull
Hn, respectively.
Our first theorem concerns the asymptotic behaviour of the expected value
of T] j as n tends to infìnity.
THEOREM
1. The expected value ofT^
<d)
En(Tiy)
=n
d-dt+t-2s-3
d 2
+ *+!
sa,tisfies
(cin^+2</+i - c 2 )(l + o(l))
for n —» oo, where c\ and c2 are constants whìch only depend on s,t,q and d
(see Appendix I).
Proof. We use òi, ò2, • • • to denote certain constants which only depend
on s,t,q and d. By defìnition,
w,_
/
T
* d L-L >-^
Encn?)
2)
..(l-\\xn\\2)"dXd(x1)...dXd(xn).
Extending to higher dimensions classical arguments due to Rényi &
Sulanke [18], [19], wecan state
*rt?= ;
/
Ud-\B
v
[~,q+l
( d ... f d \{l-W)n-d
JB
JB
+ Wn-d]rfTL1
(l-\\x1\\y...(l-\\x£)«d\d(x1)...d\d(xd),
where rj is the distance between H :— afflai,... , #</} and the tangent
hyperplane of Bd parallel to H and separated from Hn by IT, T^_t =
364
A ( ;_ 1 (conv{a:i,. . . , x ^ } ) and W the probability that a random point selected
according t o (3.1) lies in the smaller of the two parts of Bd cut off by H. Note
that W only depends on 77.
The Blaschke-Petkantschin identity of integrai geometry (see Santaló [21],
p. 201) implies that
d
2
En(T$) = (d-l)l(^
\"d-iB(~,q+i)
^ - i ^ 2
(3.2)
/ [(i - w)n~d + ^ n " V A ^ M ^ - i M ,
/
JSd~l Jo
where
(r2-P'd2)*d\d_1(x'1)...d\d_1(xd)
.
and where r = (2r) — T/ 2 )2 is the radius of the (d — l)-dimensional ball
f? r f - 1 (r) = Bd PI H. Here we have set T^-\ = Àrf_ 1 (conv{a; / 1 ,... ,x'd}) and
p1- is the distance of x1- from the centre of Bd_1(r) (j = 1 , . . . ,G?).
Some tedious but elementary calculations show that
l
J(V) = M < + 1 | d _ !
2
2
'
(2-,-V)*^
where M^ + 1 <j-i denotes the (/ + l)//i moment of the (d — l)-dimensional
volume of a (d — l)-simplex whose vertices are distributed according to
Prob(X' 6 A) =
jj-1
,
/ (l - MfY dXd_l{x')
for each Borei subset A of the (d— l)-dimensional ball with radius r and centre
0 (see Miles [15], p. 378, formulae (73) and (74)). Therefore, we have
0
J{rf) = 6,(2»? - r,2)
d2+2qd+dt-t-l
H
365
Further, it can be verifìed that
d+2g + l
r*r
(21?) - *~~
TT,^
4
/
d+2q-l
, A
Setting
d+2<?+l
2
a =
2
(d + 2j + l ) f l ( ^
+ ,
and n' = ri — d, we obtain
„
TT7Nn'
d+2 +1
/\
?
( d + 2 g - l ) ( d + 2g+1)
= (i-«i » ) (1+
4(d +
al + 3)
d+2^+3
2
""»
/ d+2?+3
( 1 + °( 1 ))J-
By Taylor's theorem we get
o d 2 +2(?d+rft-t-l
(277-77*)
2
d2+2qd+dt-t-l
= (277)
2
d2 + 2qd + dt-t4
(l
1
, XN
*? + o(rì)) •
Therefore, taking into account that Q) ~ n rf /d! , ri ~ n and VF =
o(2~ n / 2 ), expression (3.2) implies
t
t ±2
c r t . ,.„<<a // '1^1 -_-.*
¥
£„ ( T,7
a y - * W'7,_
* " 1 ( 1 L+i^..*
&3«»?~
5 , / I ~ b2n
d 2 + 2<7<J+rf* + 2 g - * - l
( 1 - 6477)77
2
6/77 .
d+2(7+l
Substituting y = ani)
2
we obtain
EniT$) ~ M ^ W ^ r ^ __iv/\, ,.„m^Sf
T^-S'r-ffl
1_67g)3+w\
^^±i+rf_ld2/
366
Finally, using that
/
i1 ~ ìTyady = r ( a + !) + ° Q ) ,0 < r < l,a > 0 ,
as n tends to infìnity (see Schneider h Wieacker [23]) and observing that the
term involving the Constant factor b5bebj does not influence the asymptotic
behaviour of En l T^ J \, we get the desired result.
A detailed proof of Theorem 1 for q = 0, that is, the case
in which the n i.i.d. random points are uniformly distributed in Bd, can be
found in [25].
REMARK.
1. The expected number offacets, surfa.ce area and volume
convex bull ofn i.i.d. random points selected according to (3.1) satisfy
COROLLARY
oftbe
(3.3)
En(f)
~ c3n*+*i+i
(3.4)
w d _ 1 - En(S)
~ c4n""^+2?+r
and
(3.5)
Kd - En(V)
~
c5n~d+Wn:
for n —> (X), where c 3? c\ and c 5 are consta,nts which only depend on q and d
(see Appendix I).
Proof
Expressions (3.3) and (3.4) follow directly from Theorem 1. In
the case of the expected volume note first that the centre of B will lie in the
convex hull Hn with high probability when n tends to infìnity. This can be
verified as follows. Consider the convex hull Hn+\ := conv(Jffn U {0}). The
centre 0 of B is not contained in Hn if and only if 0 is a vertex of Hn+\. We
bound this probability be means of a useful argument due to Bentley et al. [4]
(see also Dwyer [8]). Suppose that we have d mutually orthogonal hyperplanes
through 0. A necessary condition for 0 to be a vertex of Hn+\ is that at least
one orthant determined by the d hyperplanes is empty. Therefore, we can
state
367
< 2d\ / (1 - ||^|| 2 ) 9 rfA d (a;)] n ,
Prob (0 is vertex of Hn+1)
V
J
JR
where R = { ( z i , . . . , z^) E Md; Z{ > 0 for i = 1 , . . . , d } . It follows that
l-o(sn)
Prob (0 E Hn) =
for some £, 0 < £ < 1. Thus, if n is large enough, the expected volume of H n
can be expressed in the forai
V F£F(Hn)
)
Finally, we get expression (3.5) from Theorem
calculations.
1 after
some obvious
Our next result concerns the Minkowski quermassintegrals of the convex
hull Hn.
2. The expected vaine of the ith queimassintegral W\ ' of
the convex hull ofn i.i.d. random points selected according to (3.1) satisfìes
THEOREM
wld\Bd) - En{^vf)
2(d + 2q + S)(d-i\
/
d+2q+l
27T
V
nB
(ì4
1)!
i = 1,..., d — 1
+q+i
/
as n tends to infinity.
Proof. By definition,
\
/
„(«*>) =
E(3.6)
\
Ud-\B
( 2'«
+
n
JBd
1
JBd
/
(1 - H^ll2)» ... (1 - |M|2)»dA,K*l) . ..dXd(xn)
368
Consider the well-known integral-geometric formula for the ith. quermassintegral of a convex body K in M.d (e.g., Santaló [21], p. 222, with difFerent
notation)
WW(K)
=
(rf
du)
~ Ì ) W ° ' " ' U','~1
d-i-l
(3.7)
• -Ud-2
L
r{d-l)(T,l
JG{d,d-i)
d—v
W^-'(X'Wi-i(L"-'),
i = 1 , . . . ,rf— 1,
where G(d, d — i) is the Grassmannian of (d — i)-dimensional linear subspaces
of Md with the usuai topology and fi^-i ÌS the (unique) rotation-invariant
measure on G(d,d — i) normalized as in Santaló [21], chapter 12. Here K1
is the orthogonal projection of K onto Ld~l. Note that WQ ~1'{KI) is the
(d — i)-dimensional volume of K1.
Combining expressions (3.6) and (3.7) and applying Fubini's Theorem,
we get
\
/
En{WÌd)) =
/
d —i
d
LUQ . . . u>f"_]
Wd_i_-i . . .Wd.
[^<l + l
I
WÌd-i\H'n){l-\\x1\\'i)<'...{l-\\xn\\y
[/,-•• j
JG(d,d-i) UBd
V
"d-\B
JBd
d\d(xi)...d\d(x„)\dfi,d-i(Ld
')
and, since thera-foldintegrai in brackets does not depend on L^~l e G(d,d—i).
r(d-i
En{w\d))
= *h
+ 2\ I
d+2'
\
/
JB
d
... /
d
W{0d-l\H'n){l
\
">d-iB f ^ ' ? * 1
- \\xx\\y
. . . (1 -
n
/
\\xn\\yd\d(Xl)...d\d(xn),
JB
where H'n is the orthogonal projection of Hn onto a fìxed (d — i)-dimensional
subspace L0~l, say.
369
Decomposing the rf-dimensional volume element into orthocomplementary
lower dimensionai volume elements, we get
(3.8)
d-i + 2\ /
n
En(wld))
2
= TT5
d+2
\
/
.../ d l
JBd~l
JB ~
2'9+1
Ud-i^i
W^HB'n)K\\x[\\,...A\<\\)d^-i(x'1)...dXi.i(x'n).
Here we have set
/(||xi||,...,|K||)= / .
.../.
(al-pl)1...(ai-pl)*
d\i(x'l)...d\i(xli),
where cij = (1 — ||#',-||2)2 is the radius of the i-dimensional ball Bl(ctj) =
l
being the z'-dimensional affine space orthogonal to L0
Xj. We denote the distance of Xj from L0~l by pj (j = 1 , . . . , n).
After some obvious manipulations we get
through
n
Ui-\B ( 2 ' 9 + 1
(i-llx' 1 no...(i-iKir)
I(\\x'1\\,...,\\x'n\
Substituting into (3.8) we have
_ (d-:
En(w\d))
= Tri
t-+2\
d+2
/
\
' • /d — i i
\
/ ..../ wti)(H'n)^-\\x[\\2r^..a-\K\\2y+i
JBd~l
dX
JBd~l
d-i(x'i)---dXd-i(xn)'
n
370
Finally, we can write
r
/ < * - i +2
r (^)
l)
where EU(WQ
) is the expected volume of the convex hull of n i.i.d. random
points selected according to
Prob (X' eA) =
TTZT-i
Vf i
1
- H*'ll a )' + *«ttrf-i(*'),
for each Borei subset A of Bd~\ Thus, Theorem 2 follows immediately from
Corollary 1, expression (3.5).
The important special case of uniform distribution in the unit ball B is
stated in Corollary 2.
2. The expected vai uè of the ith quermassintegral W^ ' of
the convex hull of n i.i.d. random points selected according to the uniform
distribution in Bd satisfies
COROLLARY
d
Tl
-M
d-i
d
d + 1+
(d)
W (B ) - En{W^) ~ ?< ^ ± 4
\
2 (rf + 3)
(rf-i-1)!
» = 1,.. .,d— 1
as n tends to infinity.
The techniques used in the proofs of Theorem 1 and Theorem
2 also allow the computation of En(W^ ),i = 0,1, . . . , < / - 1, for n E N. Since
no simple explicit formulae are to be expected, we only refer to the papers of
Affentranger [1] and Buchta Sz Mùller [6], who have treated the special cases
i = 0 and i — 1, d — 1, respectively.
REMARK.
371
4. Normally distribuiteci random points.
In this section we consider the convex hull Hn of n i.i.d. random points
Xi,... ,Xn, each selected accordi ng to
Prob(X e A) = 7T~^ / exp
JA
(-\\x\\2)dXd(x)
for each Borei subset A of Md (d > 2).
For 5 G 1 + and t e N U {0}, deflne
<d}=
E
^i-i(f).
where pp is the distance of aiT F, for F € F(iir n ), from the origin. Our next
result concerns the asymptotic behaviour of Ivs J as n tends to infìnity.
THEOREM
3. The expected value ofRyJ satisfìes
*-i
r(|)
as n tends to infìnity.
The special case s = t = 0 was already studied by Rényi &,
Sulanke [18] for the piane and by Raynaud [17] for higher dimensions.
REMARK.
Proof of Theorem 3. In analogy to the proof of Theorem 1 we can write
MR$)
= ( " ) ( T r f / 2 r r f / , • • • [d[g(p)n-d
. e*P(-(\\xi\\2 + • • • +
IMI2))AJ(SI)
+
• • • d\d(?d)•>
where
1 fP
g(p) = -7=
V7? J-oo
(i-9(P))n-Vrj-x
exp(-u2)du.
372
The Blaschke-Petkantschin identity ([21], p. 201) implies that
n—dt
W
(4.1)
Jo
(1 -9(P))"-ifl — dì^Sa}psexp(-dp<)dp,
where Mt+ij-i is the (t+ l)th moment of the (d — l)-dimensional volume of
a (d — l)-simplex whose vertices are distributed according to
Prob(X' eA) = T T " ^ /
eXp
(-Wx'W^dX^x')
JA
for each Borei subset A of M d_1 (see Miles [15], p. 377, formula (70)).
We show in Appendix II that
(4.2)
/
Jo
g(p)n~dpsexp{-dp2)dp
= (d- 1)12^-^2 n - d ( l o g n ) _ : V - ( i +
0(1)).
Observing that 1 — g(p) < \ for p > 0 and that (^) ~ n /d\ we obtain the
desired result by combining expressions (4.1) and (4.2).
3. The expected number offacets, surface area and volume
of the convex hull of n i.ì.d. random points, each selected according to the
multivariate normal distribution, satisfy
COROLLARY
(4.3)
£ „ . ( / ) - -^(TTlog n ) * * 1 ,
(4.4)
En(S) ~
ud^logn)^
and
(4.5)
as n tends to infìnity.
£„(V)~/e d (logn)*
373
Proof. Expressions (4.3) and (4.4) follow directly from Theorém 3. In
the case of the expected volume it can be verifìed in the same way as in the
proof of Corollary 1 that the origin will lie in the convex hull Hn with high
probability as n tends to infìnity. Thus, the expected volume of Hn can be
expressed in the form E(V{Hn))
= -^(fljft) + o(rn) for some r, 0 < r < 1
et
and expression (4.5) eventually follows from Theorem 3.
We complete the study of the convex hull of normally distributed random
points with the following result.
4. The expected vai uè of the Uh querma,ssintegra,l W\ ' of
the convex hull of n i.i.d. random points normally distributed in Mr satisfies
THEOREM
E„{wld))
~ MOogiO^ ,
i = 1,.. „d-
1.
as n tends to infìnity.
Proof. In analogy to the proof of Theorem 2 we get
E
Ar
d- » + 2 \
d/2)n
« K') = ^ 7ihv^
r
L - L w°d~l)CK)
exp (-(libili 2 + . . . + \\xn\\2))d\d(xi)..
.d\d(xn),
where Hn is the orthogonal projection of Hn onto a fìxed (d — i)-dimensional
subspace LQ, say.
Decomposing the e?-dimensional volume elements into orthocomplementary lower dimensionai volume elements, we get
*(IM!l,..-.lk,ll)A<-!(*',)--. AÌ-,(>4) •
374
Here we have set
«•(||'«i||,...,||»'JI) = «P(-(ll*ìll 2 + . . . + ||x'J|2))
/ . . . . / . exp (-(pj + ... + pì))dXi(x'!)...dA,(*'n) =
JL\
JL'„
= (W2)"exp (-(IkìlP + --- +
where pj denotes the distance of XJ from I j ~ ' and V- is the i-dimensional
d—i
affine space orthogonal toV JL
LQ
through Xj (j = 1 , . . . , n). Therefore,
/dd-i+2\
&
i !r
Z( ¥,
H
w*)=« ' 47Ik " 'i-,•••i-, '•'"'
,(9i,
r
exp (-(\\x[\\2 + . . . + | K | | 2 ) ) A i - i ( * ì ) • • • dXd_,(x'n)
or
(d-i + 2\
.i/2 Vv / 2 >; E (wtÌ]),
En(w{d)) = *«>
n
(4.6)
r (^)
where EU{WQ ~l)) is the expected volume of the convex hull of n i.i.d. random
points selected according to
Prob(X' 6 A) = ^
f exp
(-\\x'\\2)dXd_i(xf)
JA
for each Borei subset A of E . We now obtain the desired result by
combining expressions (4.5) and (4.6).
Formulae (4.4) and (4.5) of Corollary 3 and Theorem 4 are
not surprising, in view of a result of GefFroy which states that the Hausdorffdistance between the convex hull of n normally distributed i.i.d. random
points and the rf-dimensional ball with radius (logn) 1 ' 2 centred at the origin
converges almost surely to zero (see [11], Theorem 1). It appears, however,
that our results cannot be deduced directly from Geffroy's result.
REMARK.
375
5. Ranciom points on the unit sphere
Let Hn be the convex hull of n i.i.d. random points Xi,...,Xnì
each
1
distributed uniformly on S^ . In this section we prove the following result.
5. The expected value ofthe ith quermassintegral W^ ' ofthe
convex hull ofn i.i.d. random points distributed uniformly on the unit sphere
Sd~x satisfies
THEOREM
(d2 -di + i+l\
wW(Bd)
1
K
)
- En(W&)
nK
i
)
~ ^ (—
)
2 \d + ij
d l
\
~
(d-i-iy.
/
\l=i
2?r
va.
i — 1,... ,rf — 1,
a.s n tends to infìnity.
The asymptotic behaviour of the special cases « = 0,1 and
d — 1 was already studied by Miiller [16].
REMARK.
Proof of Theorem 5. In analogy to the proof of Theorem 2 we get
d-i
d
En(W( ))
=
^/2_\
£
r
(5.1)
Jsd~l
+2
(— )
)K0)'
/
_V£
V
J
Jsd~x
where Hn is the orthogonal projection of Hn onto a fìxed (d— z)-dimensional
subspace Ld~l, say. Here <7</_i denotes (d— l)-dimensional spherical Lebesgue
measure on S _ 1 . Observing that the integrand in (5.1) only depends on the
376
(d — i)-dimensionai orthogonal projections of X\,...,
xn onto LQ *, we obtain
t+2
2x rf / 2
/ „ • - / , . W O ( '' , W.)>(II*ÌII.---MII* , .II)
JBd-%
JBd-%
[(1 - IKH2)...(1 - l l x ' J I 2 ) ] - ^ ^ ) . ..d\d_i{x'n),
where
j(ii :
n
2
''" ' [(i-lKll')...(i-IWII')] ¥ -
'(0"
Here <JÌ-\ means (i— l)-dimensional spherical Lebesgue measure on the (i— 1)dimensional sphere Sl~1(pj) = S**"1 fi Zi-, X*- being the i-dimensional affine
rf—t
space orthogonal to LQ
through Xj. We use pj to denote the radius of
Therefore, we have
n
En{W\d))
7T '
7"-
rr
d -f 2 \
d-i „
à=±
/ ì \
\
JBd-1
lBd-l
JB
or
d - i + 2'
(5.2)
#n(wf } ) = W 2
2
d+2
'-EnÒvt*),
377
where En(W0
) is the expected volume of the convex hull of n i.i.d. random
points selected according to
Prob (X1 £A) =
d — i i JA
( 2 '2
Ud-i-\B
for each Borei subset A of B
expfession (3.5) and (5.2).
l
. We obtain the desired result by combining
Appendix I: Asymptotic constants of Theorem 1 and Corollary 1
The somewhat complicated explicit asymptotic constants of Theorem 1
and Corollary 1 (see Section 3) are given in the following.
2-(,-i)r
Ci
=
(dt + 28-d-t
+\
d+2g + l
r(|)^+2,+
/rg
+f + 1 ) \ '
V
/
+
A r ftf + 2dg + dt + 2\
W -i)!T(£±2à±tl±i
r (!±A±i)/
n
J=I
r
i
\
d 2 + 2qd+d< + 2 s - < + l
<f + 2q + l
2TT
vnif+'+i
/
r(£±«Y*±»J / r g + t + i)
r(fWi)!.r(* + M' + *- t + 1 Vr f ^ + ? + i
2-(*+2)
c2 =
d-i
r
n-
i=i
'< + i +1
(d 2 + 2gc/+ c/< — * — 1)
/cP_+ 2<fy + <ft + 2s - tf + 3
(rf+2g + l)
d-f 2g+ 1
<i 2 +2dq+rf<+2gd + 2q+l
(e? + 2g - 1) /et2 + 2dq + dt + 2q + d + 2s - * + 4
d-f 2$ + l
(rf+2« + 3)
2?r
B
+ +1
v &f « .
378
c3
!r
(rf + 2g+l)rf!r
d2+2dq+l
d + 2q+l
d2 + 2dq + 2
/d 2 + 2<fy+l\
2r V d+2<j+l )
2TT
r
Q) (-
+ 2<ty + 1
5
-,U'2
\
/
2
wd_i (rf + 2rfg - 2g - 1)
2(rf-l)!
d + 2g + 3
c4 =
+4f + Ì
rf
V + 2g + l
2TT
+d
\ B G'5 +? + 1
=
u ; ^ (rf + 2 g + l ) /
2
2rf! (d + 2g + 3) \c/ + 2g + l + <*+!
/
<*+2<| + l
/
C5
<l+2 9 +l
2TT
VBl5'f+g+1.
Appendix II: Proof of expression (4.2)
Our aim is to prove that
roo
(A.l) I(n) = / g(p)n-dpsexv(-dp2)dp
./o
d+9—1
= c(</)n" d (logrc)~^~~(l + o(l))
where c(d) = (d - 1)!2 < / ~ 1 7T2. TO this end we set
(A.2)
I(n) = /i(n,e) + / 2 (n,e) +
h(n,s),
where
/,-(n,e)=/
S (p)
n
-Vexp(-rfp 2 )rfp,
»= 1,2,3,
.M,-(n,e)
and
Ai(n,e) = [0, \A°g n - e ] ,
A 3 (n, e) = [y/\og n + £ , oo]
i4 2 (n,e) = [\/log n - £ , y l o g n + £],
379
for some real number e ,
0<e<
We show in the following that
d+s-l
-d
exv(-de2)c(d)n-a(logn)-^r-(1
+ 0 (i)) <
I2(n,s)
(A3)
< exp(ds2)c(d)n-d(ìogn)à±2zl(l
(AA)
h(n,e)
+ o(l)) ,
-dt
< exp(-de2)0(n~a(\ogn)
d+s-1
2 )
and
h{n,e)
(A.5)
<
Q(n~d(\ogn)ìn).
As Ii(n,s) and h{n,e) are uniformly bounded in £, we obtain expression
(A.l) from (A.2)-(A.5) by taking the limit e -+ 0.
Proofof(A.3)
. Using the classical formula
with 0 < 0P < 1 and substituting p = \/ìog n — u/y/ìog n we obtain
exp(2w — u2/ìog n)
2y/iry/\ogn n(l — u/ìog n)
Ey/ìogW
h(n,s)
= n - d (logn)V
/
E-y/log 71
x-1
(
1-
T W—rf
0.
\
log n - 2u + :
/
\
log n/
where 0 < Qu < 1. For |^| < e y ^ S ^ ?
u 1 exp(2du —rfu2/log n)dw ,
log w
we
have
exp(—£2) < exp(—u2/ìog n) < 1 , exp(—d<s2) < exp(—du 2 /logn) < 1 ,
1 - e/ì/\ogn
< 1 — n/log n < 1 -\- e/y/ìog n and
1 - l/(log TI - 2£<v/log n) <1-0U/
(log n - 2u + - ^ — j < 1.
380
Hence
J\(ri,s) < I2(n,£) < J2(nìe) ,
(AS)
2
Ji(n,£) = exp(-<fc )
1
where
n
Vlog n
-d
_! feyfióg
(logn) 2 /
TI—d
-1
exp(2w)
12y/wn(y/\og n — e) \
n
exp(2dw)dn
log n — 2£rv^°gn
and
J2(«,e)= (1 +
n
\/log n
rf
s-l
(logrc)~2
exp(-£ 2 )exp(2u) 1 n—d
11} J. ^—*exp(2</u)<^.
l
2x/7rn(Vlog rc + e)\
1
Substituting exp(2w,) = 2v/7r(ylog n — e) I 1 —
t in
log n — 2ey/\og n
(A.6) we obtain
reVÌ^
/
J-eJte^
r
Ji(n,e) = exp(-^ 2 )2f _ 1 7r2 ( 1
n-d{\ogn)à±^1
[t+{n\l
V'iog nj
-
1V
l°g
n
— 2syJ\ogn
Ì)»-*t*-Ut
with
exp(J=2£y4o^) /
t±W
~
-ì
_
log n — 2£\/Iogn
2^(^^-e)\
Now
rt-(n)
/
Jo
f
n
1
ft-(n)
(l--) -V~ ^< /
n
JQ
d l
t ~ dt =
tÌ(n)
d
= o(l)
and
t
t i
/
(i _ ±.)« td-idt
n
Jt+(n)
f
< /
e x p ( - 0 ^ ~ 1 <ft
Jt+(n)
< const exp(-t+(n))q..(w) = o(.l),
381
where in the last inequality we have used that exp(—£)f*+1 decreases in
[/+(rc),n] for ali sufficiently large n. On the other hand we have
f'(1 _ Ì ) V - i d i = (rf _ i). + o(l)
j0
n
n
(see Schneider Sz Wieacker [23], p. 73) and hence
n
't+(n)
ft+(
) (/
/
f n\ -d n-d , ,
t\
d 1
1--
t - dt = (d-l)\(l
+ o(l))
t-(n)
Consequently
Vi(n,e) = &^(-ds2)c(d)n~d(ìogn)i±i=1(l
+ o(lj),
which proves the first inequality in (A.3). In the same way it can be verified
that
J2(n,e) = exp(de2)c(d)n-d(\ogn)à±T:1
(1 + o(l)),
which completes the proof of (A.3).
Proof of (A4) . Analogous arguments as used above show that
ry/ìogn-e
.
g(p)n-dps
Ii(n,e)<J.
exV(-dp2)dp
< constexp(—de2) I 1
(logn)
2
.
)
+ o(rn),
exp ( —2c?v/ìog n)
,l
+o(r j,
so that
(rc.£Ì < exDf—df2)
olii~d(loQ
0<r<l,
. d+s-1
f 77, 1 2
382
Proof of (A.5) . We have
roo
h(n,£)=
roo
9(p)n~dPs
/
exp(-dp2)dp
\/log n + e
ps
< /
exp(-dp2)dp
•'\Zlog
Substituting /? = Vlog n ( l -f <) in the last integrai, we conclude
h(n,e)
< 7i~ d (logn) 2
( l + J) s exp(-dlogrc* 2 )cfó
/
io
= constn-d(logn)l £
( S ) r ( ^ y i ) ( d l o g n)"2
< c o n s t a - (log n) 2
= 0(n-d(ìogn)2)
,
as required.
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Fernando AFFENTRANGER
Mathematisches Institut der Albert-Ludwigs-Universitàt,
Hebelstrasse 29, 7800 Freiburg i. Br., Germany.
Lavoro pervenuto in redazione il 9.4.1991.