  The objective of this paper is to improve the customary definition of
redundancy by providing quantitative measures in its place, which we coin upper
and lower redundancies, that match better with an intuitive understanding of
redundancy for finite frames in a Hilbert space. This motivates a carefully
chosen list of desired properties for upper and lower redundancies. The means
to achieve these properties is to consider the maximum and minimum of a
redundancy function, which is interesting in itself. The redundancy function is
defined on the sphere of the Hilbert space and measures the concentration of
frame vectors around each point. A complete characterization of functions on
the sphere which coincide with a redundancy function for some frame is given.
The upper and lower redundancies obtained from this function are shown to
satisfy all of the intuitively desirable properties. In addition, the range of
values they assume is characterized.
