Volumes of Generalized Unit Balls Author(s): Xianfu Wang

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Volumes of Generalized Unit Balls
Author(s): Xianfu Wang
Source: Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 390-395
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/30044198
Accessed: 02-06-2015 19:52 UTC
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390
MATHEMATICSMAGAZINE
of Y. Eitherway, f (f (r)) = 7r, and f is a well-defined,sign-reversinginvolution,as
desired.
I
In summary,we haveshowncombinatorially
thatforall valuesof n, therearealmost
as manyeven derangementsas odd derangementsof n elements.Or to put it another
way, when randomlychoosinga derangementwith at least five elements,the oddsof
havingan even derangementarenearlyeven.
Acknowledgment. We are indebtedto Don Rawlings for bringing this problemto our attentionand we thank
MagnhildLien, Will Murray,and the refereesfor many helpful ideas.
REFERENCES
1. R. A. Brualdi,IntroductoryCombinatorics,3rded., Prentice-Hall,New Jersey, 1999.
2. C. A. McCarthyand A. T. Benjamin,Determinantsof the tournaments,this MAGAZINE,
69 (1996), 133-135.
3. C. D. Olds, Odd and even derangements,Solution E907, Amer Math.Monthly,57 (1950), 687-688.
Volumesof GeneralizedUnit Balls
XIANFU WANG
UBCOkanagan
3333 UniversityWay,Kelowna
B.C.,Canada,V1V1V7
[email protected]
Diamonds,cylinders,squares,stars,andballs. These geometricfiguresarefamiliarto
students,butwhatcouldtheypossiblyhavein common?One answeris:
undergraduate
They aregeneralizedballs.The standardEuclideanball canbe distortedinto a variety
of strange-shapedballs by linearand nonlineartransformations.
The purposeof this
note is to give a unifiedformulafor computingthe volumesof generalizedunitballs
in n-dimensionalspaces.
A generalizedunitball in Rn is describedby the set
BPI
(1)
wherepl > 0, P2 > 0,..., Pn > 0.
Whenthe numberspi1 ...., pn areall greaterthanor equalto 1, the unitballB
is convex. Since IxlPis not concaveon [-1, 1] for 0 < p < 1, BP,...Pn is not necesPn,= p > 1, we obtain
sarilyconvex anymorewhen n > 1. Whenp1 = P2 = ...=
the usual1, ball. The 12 ball is denotedby B. By choosingdifferentnumberspi, we
can alterthe appearanceof the generalizedballs greatly,as shownin FIGURE1 with
examplesin R3
Motivatedby an articleby Folland[5], I deriveda unifiedformulafor calculating
the volumeof these balls. Althoughthe volumeformulasfor the standardEuclidean
ball TBand simplex have been knownfor a long time [4, pp. 208, 220], the unified
formulais (relatively)new. It is surprisingthatno matterhow strangethe balls look,
the volumeof anyball canbe computedby a single formula,as follows:
THEOREM. Assume pi, ..., pn > O. The volume of the unit ball Bp
is equal to
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391
VOL. 78, NO. 5, DECEMBER2005
-1
-1
-0.5
-0.5
z 0
z 0
0.5
0.5
1
1
-0.5
0
x
0.5
7-0.5
0y
0.5
11
1
-1'
-0.5
-1
'-1
'0.5
x
'0
0
'0.5
0.5
1
y
1
(b)
(a)
1
0.5
z 0
-0.5
-1
-1
1
-0.5
0.5
0
y
0.5-
1
'0.5
Ux
(c)
Figure1
(a)
|x3|
(2)
2n
The volume of the positive orthant part, where all x-values are positive, may be obtained by removing thefactor of 2nfrom theformula.
The formulainvolvesthe gammafunction,whichwe reviewfor readerswho may be
unfamiliarwithit. For0 < t < oo, we define
F
The integralconvergesfor t > 0. The followingfacts will be needed:For u > 0 and
v > 0, we have
v
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(3)
392
MAGAZINE
MATHEMATICS
and
(4)
su-I'(I
Althoughthe integralin F(t) becomesinfinitefor t < 0, (3) providesan analyticcontinuationformulato defineF(t) for t < 0. The functionF has discontinuitiesonly at
t = 0, -1, -2,.... Moredetailscanbe foundin Folland[6, pp. 344-346].
Proof
Step 1. We begin withthe fact that
.Pn
andapplya changeof variablesthatdeformsthe generalizedball intoBI,the standard
ball:Let pn/2.
Forthe function
((y)
the Jacobiandeterminantis
Pn
Readersmay consultFolland[6, p. 432] for a detailedproofof the changeof variablesformula,whichis ournext ingredient.We use it to obtain
Pn
Step2. Assumea1,..., an,> -1. We claim:
(5)
f
(ai + 1)/2 for i = 1, ..., n.
To verify this claim,we developa recursionformula.Let I (a, ..., a,) denotethe
integralin (5). Wethenevaluatethis as aniteratedintegralstartingwithxl as outermost
variable.
where Pi :=
-1
The innerintegrationtakesplace over a ball of radiusr =
ables again, we set (x2,... , x)
=
r(y2, ...,
y,) to get
1-
x.
Changingvari-
dxn
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VOL.78, NO. 5, DECEMBER
2005
This gives I (a, ...,
393
an) =
=F
Henceby (4),
i
This provides a recursion formula connecting I(al, a2, . . . , an) and I (a2, ... , an).
Applyingthe recursionformula(n - 1) times,aftercancellation,we obtain
(6)
I(a
But
I(a
Puttingthis into (6) yields (5).
Step 3. When ai = 2/pi - 1 for i = 1, ..., n, (5) gives
I(a
Hence
I(a
The volumeof positiveorthantpartfollows fromtherebeing 2n orthantsin Rn.
I
In (1), you might arguethat pi cannotbe infinite,but, my dearreaders,we can
consider a limiting case. Let us write
I(a
We proceed to single out a few special cases (calculus students' delights):
1. Some pi = +oo: as F is continuous on (0, +oo), we have V
) =
(B~Pl...1P
I(a
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394
MATHEMATICSMAGAZINE
= +oo, the volume of the ball is 2n, and the
In particular,when pi = P2 = Pn,
is
an
n-dimensional
shape
hypercube (excluding the portions of its boundary where
two or more xis are simultaneously 1). When pi = P2 = 2, p3 = oo, the generalized
ball is a circular cylinder in R3.
=
2. When p = p2 = ... = Pn = p > 0, we have
V(BI,...p)
["(n/p
=
Recall that IF(1) 1, F1(1/2) = r1l/2,and r1(n) = (n - 1)!. For p = 2, the generalized ball is the standardEuclidean ball with volume 27rn/2/(nIF(n/2)). For p = 1, the
generalized ball
1),
is an n-dimensional diamond, and has volume 2n/n!. For p = 1/2, the generalized
ball has volume 22n/(2n)!, and its shape is an n-dimensional star.These are two of the
balls shown in FIGURE1.
Surprisingly, for 0 < p < oo we find the n-dimensional ball has smaller volume
when n becomes larger, and that
terms
Here we use Stirling's formula: 1-(x) ~ V2-rxx-1/2e-x, where - means that the ratio
of the quantity on the left and right approaches 1 as x -+ oo [6, p. 353].
3. For the ellipsoid {(x1,
with
>
a
linear
transformation
reduces
it
to
in
the
form
and
the
theorem
(1)
0,
simple
ai
yields its volume as
a
FIGURE2 shows two ellipsoids in 1R3to give readers an idea of their appearance.
-1
-0.06
-05
z
z 0.
0.5
1
-0.1
-0.05
0
y
0.05
(a)
0.1
10.4
0.2
LO2 x
OA-
0
0.06
1
0.1
-0.5
y
0
0.5
1
0.2
X0.10 x
-0.2
(b)
<1
+ Ix31< 1; (b) 161x12+ Ix212+ 41x311/2
Figure2 (a)161xl14+ 31x211/2
Remark After I obtained this result, Dr. J. M. Borwein, at Simon Fraser University, informed me that the 19th-centuryFrench mathematicianDirichlet had obtained
a similar result using a different method [3, pp. 153-159]. An induction-freeproof to
the volume formula of the lp,ball, via the Laplace transform,has been given by Bor-
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395
VOL. 78, NO. 5, DECEMBER2005
wein andBaileyin [2, pp. 195-197]. Similarly,one canderivean induction-freeproof
to the volumeformulaof generalizedballs (2) using the Laplacetransform.Finally,
we remarkthat more propertieson the gamma function and volume of Euclidean balls
canbe foundin Stromberg[7, pp. 394-395].
Acknowledgment. This work was supportedby NSERC and by the GrantIn Aid of OkanaganUniversityCollege.
REFERENCES
1. J. A. Baker,Integrationover spheresand the divergencetheoremfor balls, Amer Math.Monthly104 (1997),
36-47.
2. J. M. Borwein, D. H. Bailey, Mathematicsby Experiment:Plausible Reasoning in the 21st Century,A.K.
Peters,2003.
3. J. Edwards,A Treatiseon the IntegralCalculus,Vol. II, Chelsea PublishingCompany,New York, 1922.
4. W. Fleming, Functionsof Several Variables,Springer-Verlag,New York, 1977.
5. G. B. Folland,How to integratea polynomialover a sphere,Amer Math.Monthly108 (2001), 446-448.
6. , AdvancedCalculus,PrenticeHall, UpperSaddle River,NJ, 2002.
7. K. R. Stromberg,An Introductionto Classical Real Analysis, WadsworthInternationalMathematicsSeries,
1981.
ProofWithoutWords:A TriangularSum
t(n)=
k=O
3t(1)
3t(2)
3t(4)
3t(211)
k=O
Exercises: (a)
L
(b)
t
-ROGER
B. NELSEN
LEWIS& CLARKCOLLEGE
PORTLANDOR 97219
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