  This manuscript is a preliminary pre-print version of a journal submission by
the authors, revisiting the problem of range measurement based localization of
a signal source or a sensor. The major geometric difficulty of the problem
comes from the non-convex structure of optimization tasks associated with range
measurements, noting that the set of source locations corresponding to a
certain distance measurement by a fixed point sensor is non-convex both in two
and three dimensions. Differently from various recent approaches to this
localization problem, all starting with a non-convex geometric minimization
problem and attempting to devise methods to compensate the non-convexity
effects, we suggest a geometric strategy to compose a convex minimization
problem first, that is equivalent to the initial non-convex problem, at least
in noise-free measurement cases. Once the convex equivalent problem is formed,
a wide variety of convex minimization algorithms can be applied. The paper also
suggests a gradient based localization algorithm utilizing the introduced
convex cost function for localization. Furthermore, the effects of measurement
noises are briefly discussed. The design, analysis, and discussions are
supported by a set of numerical simulations.
