On the Perimeter of a Triangle in a Minkowski Plane
Author(s): H. Maehara and T. Zamfirescu
Source: The American Mathematical Monthly, Vol. 112, No. 6 (Jun. - Jul., 2005), pp. 521-522
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/30037523
Accessed: 25/08/2009 09:03
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5. M. Protterand H. Weinberger,MaximumPrinciples in Differential Equations, Prentice Hall, Englewood
Cliffs, NJ, 1967.
6. A. Zygmund, TrigonometricSeries, 2 vols., CambridgeUniversity Press, Cambridge, 1959.
MathematicsDepartment,Northwestern University,Evanston, IL 60208-2730
[email protected]
On the Perimeter of a Triangle
in a Minkowski Plane
H. Maehara and T. Zamfirescu
A Minkowskiplane is a planeendowedwith a normthatis not necessarilyEuclidean.
The Minkowskilengthof a curvein a Minkowskiplaneis definedin a manneranalogous to the way it is definedin the Euclideancase. A circleof radiusr in a Minkowski
plane with norm I|| IIis the locus of points X satisfying IX - P II= r for a fixed
"center"P. It is a convex curvethatis symmetricwith respectto its center.Golab's
theoremshowsthattheMinkowskilengthof a circleof unitradiuslies between6 and8
(see [2]). Let L denotethe Minkowskilength.We establishthe following:
Theorem 1. Let ABC be a triangle in a Minkowskiplane, and suppose that one is the
minimum radius of a circle that encloses the triangle ABC. Then 4 < L(ABC) < 6.
Since thereis an arbitrarilythin trianglewith two sides close to a diameterof the
circle,the lowerbound4 cannotbe improvedin any Minkowskiplane.Forthe case of
the Euclideannormit is not difficultto see thatthe upperbound6 can be reducedto
3/3 [1]. However,in the max-normIl(x,y)IIo = max{Ixl,lyl} any threecomers of
the unit circle (whichis a Euclideansquare)form a trianglewith perimeter6. Hence
the upperboundof the theoremcannotbe improvedin general.
Proof Let r be a circle circumscribedaboutABC in the sense that it has smallest
radiusamongall circlesthatcontainA, B, andC in theirconvexhulls.It is easily seen
thateitherthe centerof the circle is inside the triangleor on one of its sides, or there
exists a congruentcircumscribedcirclewith this property.So we may assumethatthe
centerof F is the origin O andlies insideABCor on one of its sides. The circle r is
supposed to have radius 1. Since I||A- BII < 2, IIB - CII < 2, and |IC - AI| < 2, we
have L(ABC) < 6.
We have to show that L(ABC) _ 4. If I||A- BII = 2, then
||A - BII+ II|B- C|| + I||C- Al||
A - BII+ IIB- Al = 4.
Hencewe considerthe case that ||A - BII,IIB - CII,and ||C - All areall less than2.
Notice thatin this case O lies insideABC,while A, B, andC all lie on r.
Let D, E, F, G, H, and I be points on the sides of ABC such that ADOI, BFOE,
andCHOGareparallelograms
(see Figure1). Then
II||A
- D|| + ||A -
= IIA- DII+ ||D||
||AII= 1,
||B - E| + ||B - F|| = |B - E| + \\E\\ _ \B\ = 1,
||C - G|| + ||C - H|| = ||C - G|| + ||G|| > ||C|| = 1.
June-July 2005]
NOTES
521
B
F
G
C
Figure 1.
To provethatL(ABC)> 4, it is enoughto showthat
||D-E||+|IIF-GII
+ IIH-I
> 1.
Amongthe similartrianglesDEO, OFG, andIOH, we may supposethatDEO is the
biggestandIOHthe smallest.Then ID - Eli IIF|| _ I|I. Let J be the pointsuch
thatOFGJis a parallelogram,and let K be the intersectionof GJ andAC. Then the
triangleKGCis a translateof IOH,and
II||D- E||
IIF||
II= J - | GII> IIJ - KII,
lIF - Gll = IJII,
IIH- III||
= ||IIC
- K||II.
Therefore,
II|D
- EI||+ IIF- GII+ IIH- I II> IIJ- KII+ IIJ||||II
+ IIC- KII
> IICII=1.
U
Remark. The lowerboundin the theoremis valid for any closed curveinsteadof a
triangle.Moreprecisely,let i2be a simpleclosed curvein a Minkowskiplane,andlet
one be the smallestradiusof a circlethatencloses 2. ThenL(2) > 4.
ACKNOWLEDGMENT. The paper was startedwhile the second authorvisited Ryukyu University in 2002,
where he was supportedby the JSPS InvitationFellowship Programfor Research in Japan.
REFERENCES
1. H. Maehara,Onthetotaledge-lengthof a tetrahedron,
thisMONTHLY
108 (2001)237-243.
2.
A. C. Thompson, MinkowskiGeometry,CambridgeUniversity Press, New York 1996.
College of Education, RyukyuUniversity,Nishihara, Okinawa, 903-0213, Japan
[email protected]
FachbereichMathematik,Universitit Dortmund,44221 Dortmund,Germany
[email protected]
522
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