Abstract. Given [Formula: see text] points [Formula: see text] on the unit circle in [Formula: see text] and a number [Formula: see text], we investigate the minimizers of the functional [Formula: see text]. While it is known that each of these minimizers is a spanning set for [Formula: see text], less is known about their number as a function of [Formula: see text] and [Formula: see text] especially for relatively small [Formula: see text]. In this paper we show that there is unique minimum for this functional for all [Formula: see text] and all odd [Formula: see text]. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for [Formula: see text] odd, there exists a sequence of points [Formula: see text] so that a unique (up to some isometries) minimizer exists on each of the subintervals [Formula: see text].

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.