Acta Math. Hungar. 67 (1-2) (1995), 123-129.

S U M S OF D I S T A N C E S

IN NORMED

SPACES

M. GHANDEHARI (Monterey)

Let X be a real normed linear space. For each finite subset { X l , . . . , . . . , x~} C X let s = S ( X l , . . . , xr) denote the sum of all distances determined by pairs from { X l , . . . , x~}. T h a t is, let

(1)

8(Xl,"" .,Xr) = ~

[Ixi- XjII,

where the sum is taken over all integers i , j , satisfying 1 < i < j < r. Let S = { x: I[x[[ = 1} be the unit sphere of X. Martelli and Busenberg [8] use inequalities in connection with work on a u t o n o m o u s systems of differential equations to prove the following theorem. THEOREM 1. Let X l , . . . , x~ be r points on the unit sphere S of a normed space. Assume that the convex hull of x l , . . . , x r is at distance d from the origin measured with respect to the norm. Then

(2)

>=2(r- 1)(1-d).

To prove T h e o r e m 1 we use the following theorem which was conjectured by Griinbanm and proved in [1]. THEOREM 2. Let X l , . . . ,x~ be points in a real normed linear space X . Suppose p belongs to the convex hull of { X l , . . . , xr}. Then

(3)

s ( x l , . . . ,xT) > (2r - 2)rain I [ x i - PlI,

where the m i n i m u m is taken over all i satisfying 1 < i < r. PROOF OF THEOREM 1. There is a point p with distance d from the origin which belongs to the convex hull of { X l , . . . , xT}. There is an integer j , l _ _ < j < r, such t h a t m i n [ [ x i - p ] l = I]xj-p]l" By Theorem 2 and the triangle inequality

s(xl,...,x~) > 2 ( r - 1)min [[x~- pll = 2 ( r - 1)J[xj-p[[ > 2 ( r - 1)(1 - d), where the last inequality is obtained by applying the triangle inequality to a triangle with vertices p, xj and the origin. Thus the proof of T h e o r e m 1 is complete. [] 0236-5294/95/$4.00 (~) 1995 Akad6miai Kiad6, Budapest

124

M. GHANDEHARI

In the following we review results related to the inequality (2). Consider r points x l , x 2 , . . . ,xr, in a real normed linear space X with norm ]]. H. The convex hull of midpoints of line segments joining xi and xj for all i and j , i ~ j , is called the midpoint polyhedron for x ~ , . . . , x~. Chakerian and the author [3] proved the following. THEOREM 3. Let p belong to the midpoint polyhedron of C X . Then

{Xl,... ,Xr} C

?,

(4)

(2r-- 2) E l l p - - Xil i 5

rS(Xl,''',Xr)"

i=l

As a consequence of the above the following is shown in [3]. THEOREM 4. Let x l , . . . , x ~ be points on the unit sphere S o f a normed linear space X , and suppose that the origin o belongs to the convex hull of { x l , . . . , x ~ } . Then

(5)

s ( x l , . . . , x r ) > 2r - 2.

Theorem 4 is due to Chakerian and Klamkin [4], which they proved for Euclidean spaces and for the Minkowski plane. Wolfe [10] proved Theorem 3 using the concept of metric dependence. :)C 2

~3 J' " ' ~ ~Cr

351

Fig. 1 Equality for Theorem 3 Figures 1 and 2 give examples where equalities are attained in Theorems 3 and 4. In the remainder of this article we use techniques from integral geometry to prove special cases of Theorem 2 in two and three-dimensional Minkowski spaces. Minkowski spaces are simply finite dimensional normed linear spaces. Smoothness assumptions on the boundary of the unit disk E for a Minkowski plane will enable us to use Crofton's simplest formula from integral geometry to give a proof of (4) for three points {xl,x2,x3}. Acta Mathematica Hungarica 67, 1995

SUMS OF DISTANCES IN NORMED SPACES

125

322

"~C.3 ) .

9 ~C r

3C~

Fig. 2 Equality for Theorem 4 If the unit ball for a 3-dimensional Minkowski space is a zonoid, then we use integral geometry to prove (4) for the case of four points xl,-x2, x3, and x4 forming a simplex. A zonoid is a limit of sums of segments. Bolker [2] discusses equivalent conditions for a convex subset of R ~ to be a zonoid. Santal6 [9] is a good reference for integral geometry in the Eucfidean spaces. Given a curve C in the Eucfidean plane, let L denote the length of C. Crofton's simplest formula is

/ ff ndpdO= 2L

(6)

where the integral is taken over all lines intersecting C, the pair (p, 8) is the polar coordinate representation of the foot of perpendicular from the origin to the line, and n is the number of intersections of a line with coordinates (p, O) with C. The differential element dG = dp dO is the integral geometric

density for lines. Chakerian [5] treats integral geometry in the Minkowski plane. We sketch the definitions he uses to develop Crofton's simplest formula in the Minkowski plane. Assume the unit circle E is "sufficiently" differentiable and has positive finite curvature everywhere. Parametrize E by twice its vectorial area r and write the equation of E as

t=t(~), E is called the representation

0__
Iltll=llt-Oll=l.

indicatrix. Define the isoperimetrix T by the parametric

n(r

dt(r de'

0 < r < 2~r. = = Acta Mathematica Hungarica 67, 1995

126

M. GHANDEHARI

Define A(r by ~ = -A-l(r162 Then the density for lines in twodimensional Minkowski spaces is defined as follows. Let G = G(p, r be parallel to the direction t(r The equation of G is [t(r

x] = p ,

where Ix, y] = xly2 - x2yl. Then the density dG for lines is

dG = ~-1(r dpdr It is then shown in Chakerian [5] that the simplest formula of Crofton holds:

(7)

/ n dG = 2i,

where n is t h e number of intersections of a line G with a curve C, integration is taken over all lines intersecting C and t in the Minkowskian length of C. We use Crofton~s simplest formula to prove the following corollary of Theorem 3. Recall that we defined the midpoint polyhedron of r points earlier. In the case of three points the midpoint polyhedron is called the midpoint triangle. COROLLARY 1. Consider a point p in the midpoint triangle of a triangle with vertices xl,x2, and x3. Then 3

(s)

lip-xill < 3 i----1

l_
INTEGRAL GEOMETRIC PROOF. Let ~i,i = 1,2,3 be the fine segment joining p to xi. Let ~i = [[P- xi[[. Let #i be the measure of lines which intersect L:i only. Assume #ij is the measure of the fines which intersect 1:i and s and let ~(T) denote the length of the triangle with vertices xl, x2, x3. Then /(T) = ]21 + ]22 + ]23 + ]212 + ]223 + ]2~1 = ~- ]21 21-~12 -~ ]213 + ]22 -~ ]221 -[- ]223 '~ (J23 - ]212). Hence ~ ( T ) - 2~1 + 2~2 -~- (]23 - ]212).

Similarly, ~ ( T ) -- 2~ 2 -~- 2~ 3 --~ (]21 - ]223), Acta Mathematica Hungarica 67, 1995

SUMS OF DISTANCESIN NORMED SPACES

127

and g(T) = 291 -1- 293 + (#2 - .13)Adding the last three inequalities we obtain

3g(T) = 4(el + g2 + g3) + ( P 3 - .12) -1-(Pl - P23) -~-(#2 - #13) ~ 4(gl + g2 + g3) since ( . 3 -- .12) ~ 0, (.1 -- .23) ~ 0, (.2 -- .13) ~ 0. To prove, for example, that #a => ,23, we reflect s through p and notice that any fine which intersects s and l;3 will intersect the reflection of s but there are lines which intersect the reflection of s and miss s and s We are using the fact that the measure of the lines which intersect the reflection of s only have the same measure as the lines which intersect s only. Note that equality holds if and only if reflection of s will coincide with t;2 and s [] As a consequence of the above we obtain the following result of Laugwitz

[7]: COROLLARY 2. A triangle inscribed in the unit circle of a Minkowski plane and having the center as an interior point has perimeter greater than 4. For curves in three dimensional Euclidean spaces, the integral geometric analogue of Crofton's simplest formula is (9)

/ / f n(O,r

zrL

where n(O,r is the number of intersections of a plane of coordinates (0, r p) with the curve C and integration is taken over all planes intersecting C. See Santal6 [9]. For the case where the unit ball is a zonoid, Chakerian [6], Appendix, gives the analogue of (9) for a Minkowski space. With this in mind we sketch a proof of the following special case of Theorem 2 (see Figure 3). COROLLARY 3. Consider a tetrahedron with vertices xa, x2, x3, and x4 in a three-dimensional Minkowski space. Let p be a point in the midpoint polyhedron. Then 4

(10) i=1

2 l=
PROOF. Denote the line segment joining p to xi by s and let gi = l i p - xill. Let , i be the measure of planes intersecting/2i only. Suppose #ij is the measure of planes intersecting s and s only and similarly define , i j . Then, 291 = 2.1 -4- 2,12 + 2.13 -t- 2.14 -4- 2.124 + 2.134 -4- 2.123, Acta Mathematica Hungarica 67, 1995

128

M. GHANDEHARI

351

~2

~4 Fig. 3 For inequality (10) 2~2 = 2.2 qu 2.21 + 2.23 ~- 2.24 + 2.213 + 2.214 + 2.234, 2~3 = 2.3 + 2.31 + 2.32 + 2.34 + 2.314 + 2.324 + 2.321. The sum of the edge length of the tetrahedron is denoted by L(T) and is given by g(T)

= 3.a + 3.2 -}- 3.3 + 3.123 + 3.124 + 3.134 + 3.234+ +4[.12 + .23 + .34 + #1~ + .14 + .24].

The expression in brackets is multiplied by 4 since any fine intersecting/:i a n d / : j intersects the tetrahedron in 4 points. Hence ~.(T) - 2(gl + g2 + ~3) -- (.1 - .234) -{- (.2 - .134) -~- (.3 - .124)+ +3(.4 -- "123) -~- 2("34 -V "14 -~-"24). But using reflection (#i - .jk~) > 0, i ~ j, k, ~. Hence g(T) >__2(~ + ~2 + ~3)Similarly g(T) >= 2(g2 + ~3 + g4), s => 2(gl + g3 + g4) and g(T) _>_2(~1 + + g2 +/4) which yields 4~(T) >= 6(~1 + ~2 + ~3 + Q), proving (10). [] Acta Mathematica Hungarica 67, 1995

SUMS OF DISTANCES IN NORMED SPACES

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References [1] A. D. Andrew and M. A. Ghandehari, An inequality for a sum of distances, Congressus Numerantium, 50 (1985), 31-35. [2] E. D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc., 145 (1969), 323-345. [3] G. D. Chakerian and M. A. Ghandehari, The sum of distances determined by points on a sphere, Annals of the New York Academy of Sciences, Discrete Geometry and Convexity, 440 (1985), 88-91. [4] G. D. Chakerian and M. S. Klamkin, Inequalities for sums of distances, Amer. Math. Monthly, 80 (1973), 1009-1017. [5] G. D. Chakerian, Integral geometry in the Minkowski plane, Duke Math. Jour., 29

(1962), 375-382. [6] G. D. Chakerian, Integral Geometry in the Minkowski Plane, Ph.D. thesis, University of California (Berkeley, 1960), Appendix. [7] D. Langwitz, Konvexe Mittelpunktsbereiche und normierte Rs Math. Z:., 61

(1954), 235-244. [8] M. Martelli and S. Busenberg, Periods of Lipschitz functions and lengths of closed curves, Proc. Intl. Conf. on Theory and Application of Differential Equations, Ohio University (1988), pp. 183-188. [9] S. L. A. Santal6, Introduction to Integral Geometry, Hermann (Paxis, 1953). [10] D. Wolfe, Metric dependence and a sum of distances, the geometry of metric and linear spaces, Proc. Conf. Michigan State Univ., (East Lansing, Mich., 1974), pp. 206-211. Lecture notes in math. vol. 490, Springer (Berlin, 1975).

(Received October 6, 1993) DEPARTMENT OF MATHEMATICS NAVAL POSTGRADUATE SCHOOL MONTEREY, C A L I F O R N I A 93943 U.S.A.

Acta Mathematica Hungarica 67, 1995