Scale-invariant density curvature ($D(f)=\sigma (f{)}^{5}R(f^{\prime \prime })$)

For $f_a(x)=af(ax)$ with a positive, differentiation and change of variables give $R(f_a^{\prime \prime })=a^{5}R(f^{\prime \prime })$, while its standard deviation is $\sigma (f)/a$. Their product D is therefore invariant under scale changes, and translation also leaves it unchanged. Raw integrated squared curvature can be arbitrarily small under dilation even though this dimensionless shape measure is unchanged.