Measuring cubeness in the limit cases

In this paper we show that the recently introduced family of the cubeness measures Cβ(S)(β>0) satisfy the following desirable property: limβ→∞Cβ(S)=0, for any given 3D shape S different from a cube. The result implies that the behaviour of cubeness measures changes depending on the selected value of β and the cubeness measure can be arbitrarily close to zero for a suitably large value of β. This also implies that for a suitable value of β, the measure Cβ(S) can be used for detecting small deviations of a shape from a perfect cube. Some examples are given to illustrate these properties.