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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 REFERENCES Linked references are available on JSTOR for this article: http://www.jstor.org/stable/30044198?seq=1&cid=pdf-reference#references_tab_contents You may need to log in to JSTOR to access the linked references. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Magazine. http://www.jstor.org This content downloaded from 165.91.112.105 on Tue, 02 Jun 2015 19:52:30 UTC All use subject to JSTOR Terms and Conditions 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 This content downloaded from 165.91.112.105 on Tue, 02 Jun 2015 19:52:30 UTC All use subject to JSTOR Terms and Conditions 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 This content downloaded from 165.91.112.105 on Tue, 02 Jun 2015 19:52:30 UTC All use subject to JSTOR Terms and Conditions (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 This content downloaded from 165.91.112.105 on Tue, 02 Jun 2015 19:52:30 UTC All use subject to JSTOR Terms and Conditions 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 This content downloaded from 165.91.112.105 on Tue, 02 Jun 2015 19:52:30 UTC All use subject to JSTOR Terms and Conditions 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- This content downloaded from 165.91.112.105 on Tue, 02 Jun 2015 19:52:30 UTC All use subject to JSTOR Terms and Conditions 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 This content downloaded from 165.91.112.105 on Tue, 02 Jun 2015 19:52:30 UTC All use subject to JSTOR Terms and Conditions