A point-determination problem in normed spaces

Gyorgy Pal GeherUniversity of Szeged, Hungarygehergy@math.u-szeged.hu

Abstract

It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points{p0, . . . pd} in Rd, then the Euclidean distances {|x − pj|}dj=0 determine the point x in Rd (andtherefore in the simplex Conv(p0, . . . pd)) uniquely. Here we investigate a similar problem in gen-eral normed spaces. Namely, we present a characterization of those, at least d-dimensional, realnormed spaces (X, ‖ · ‖) for which every set of d+1 affine independent points {p0, . . . pd} ⊂ X ,the distances {‖x−pj‖}dj=0 determine the point x lying in the simplex Conv(p0, . . . pd) uniquely.

IntroductionLet (X, ‖ · ‖) be a real normed space. The subset P ⊆ X is a resolving set for S ⊆ X if

x1, x2 ∈ S, ‖p− x1‖ = ‖p− x2‖ (p ∈ P ) =⇒ x1 = x2.

This notion was originally defined in general metric spaces in• L.M. Blumenthal, Theory and Applications of Distance Geometry, Clarendon Press, Oxford, 1953.

Then in 1975, a very similar concept was introduced in graph theory. Because of applications ininformatics, robotics, biology and chemistry, this notion has been being investigated.

The notion of resolving sets in metric spaces is naturally related to the characterization of theirisometries. Recently, motivated by some problems in quantum mechanics, this notion was im-plicitly used in the following papers in order to describe isometries of certain classes of matrices oroperators:• L. Molnar and W. Timmermann, Maps on quantum states preserving the Jensen-Shannon diver-

gence, J. Phys. A: Math. Theor. 42 (2009), Article ID 015301, 9 pp.• L. Molnar and G. Nagy, Isometries and relative entropy preserving maps on density operators,

Linear Multilinear Algebra 60 (2012), 93–108.•G. Nagy, Isometries on positive operators of unit norm, Publ. Math. (Debrecen) 82 (2013), 183–

192.•Gy.P. Geher, An elementary proof for the non-bijective version of Wigner’s theorem, Phys. Lett. A,

378 (2014), 2054–2057.Also recently, motivated by complex analysis, some basic results on resolving sets in general metric

spaces were provided in• S. Bau and A.F. Beardon, The metric dimension of metric spaces, Comput. Methods Funct. Theory

13 (2013), 295–305.

The Main GoalFor a number d ∈ N, d ≥ 2, we say that X with dimX ≥ d has the property (SRSd), if it satisfies thefollowing condition:

every set {p0, p1, . . . pd} of d + 1 affine independent points resolves its convex hull. (SRSd)

Our goal is to characterize real normed spaces which satisfy (SRSd).

p0

p1

p2

x1x2

p0

p1

p2

p3

x1

x2

This cannot happen in an (SRS2) or (SRS3) normed space, respectively.Here if the colors are the same, then the distances between the corresponding points are equal.

Our approach to the problemWe shall make use of the following notation: for every two points x1, x2 ∈ X, x1 6= x2 let

B(x1, x2) := {z ∈ X : ‖z − x1‖ = ‖z − x2‖} ⊆ X,

which is usually called the bisector of x1 and x2.

B(x1, x2)

x1

x2

A bisector in a two-dimensional normed space lies always in a region bounded by two parallel lines.A similar statement is false in higher dimensions.

We note that geometric properties of bisectors in finite dimensional normed spaces yield variousdeep characterizations of special normed spaces. This notion is also naturally related to the studyof Voronoi diagrams. Of course in a Euclidean space a bisector is always a hyperplane. If X isa strictly convex d-dimensional space, then all bisectors are homeomorphic images of hyperplanes.See e.g. the following papers for more details:

•H. Martini, K.J. Swanepoel and G. Weifi, The geometry of Minkowski spaces - a survey part I.,Expo. Math. 19 (2001), 97–142.

•H. Martini and K.J. Swanepoel, The geometry of Minkowski spaces - a survey part II., Expo. Math.22 (2004), 93–144.

A useful re-phrasing of the problem

The normed space X with dimX ≥ d satisfies (SRSd)⇐⇒

there is no bisector B(x1, x2) with d + 1 affine independent pointsp0, p1, . . . pd ∈ B(x1, x2) such that we have x1, x2 ∈ Conv(p0, p1, . . . pd).

Main results

Surprisingly, the characterization of (SRSd) normed spaces is different in the d = 2 and d ≥ 3 cases.These characterization reads as follows:

Theorem 1.Let X be a real normed space with dimX ≥ 2. ThenX satisfies property (SRS2) ⇐⇒ X is strictly convex.

Theorem 2.Let d ≥ 3, and X be a real normed space with dimX ≥ d. ThenX satisfies property (SRSd) ⇐⇒ X is an inner product space.

Some consequences of the main results

The following is a theorem of G.K. Kalisch and E.G. Straus:

Corollary 1. Let d ≥ 2. A d-dimensional normed space is Euclidean if and only if every set of d+1affine independent points is a resolving set for the whole space.

The next two corollaries are re-phrasing of the above two theorems.

Corollary 2. Let m ≥ 2, and K be a convex, compact body in Rm with non-empty interior suchthat K = −K. The following two conditions are equivalent:

(i) for every two numbers λ1, λ2 ∈ (0,∞), and linearly independent vectors v1, v2 ∈ Rm, the inter-section

Conv(0, v1, v2) ∩ (∂K) ∩ (v1 + λ1 · ∂K) ∩ (v2 + λ2 · ∂K)

has at most one element;

(ii)K is strictly convex.

Corollary 3. Let m ≥ d ≥ 3, and K be a convex, compact body in Rm with non-empty interiorsuch that K = −K. The following two conditions are equivalent:

(i) for every d numbers λ1, . . . λd ∈ (0,∞), and linearly independent vectors v1, . . . vd ∈ Rm, theintersection

Conv(0, v1, . . . vd) ∩ (∂K) ∩(∩dj=1(vj + λj · ∂K)

)contains at most one element,

(ii)K is an ellipsoid.

References

[1] Gy.P. Geher, Is it possible to determine a point lying in a simplex if we know the distances fromthe vertices?, submitted for publication.

Acknowledgements

The author emphasizes his thanks to Gergo Nagy (University of Debrecen) who posed this problemin a personal conversation.The author was supported by the ”Lendulet” Program (LP2012-46/2012) of the Hungarian Academyof Sciences.