Is it possible to determine a point lying in a simplex if we know the distances from the vertices?

Gehér, G. P. (2016) Is it possible to determine a point lying in a simplex if we know the distances from the vertices? Journal of Mathematical Analysis and Applications, 439 (2). pp. 651-663. ISSN 0022-247X doi: 10.1016/j.jmaa.2016.03.024

Abstract/Summary

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 |}d j=0 determine the point x in Rd uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d-dimensional, real normed spaces (X, · ) for which every set of d + 1 affine independent points {p0, ...,pd} ⊂ X, the distances {x − pj}d j=0 determine the point x lying in the simplex Conv({p0, ...,pd}) uniquely. If d = 2, then this condition is equivalent to strict convexity, but if d > 2, then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these (Theorem 1) is re-proven using the fundamental theorem of projective geometry.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/81507
Identification Number/DOI	10.1016/j.jmaa.2016.03.024
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Elsevier
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