Scale-invariant density curvature
= Scale-invariant density curvature
{title2=$D(f)=\sigma(f)^5R(f'')$}
For $f_a(x)=af(ax)$ with a positive, differentiation and change of variables give $R(f_a'')=a^5R(f'')$, 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.