For this question, I am interested to find a relation between the diameter and the Hausdorff dimension of a n-dimensional interval $I$. (https://en.wikipedia.org/wiki/Hausdorff_measure)
Lets say $v(I)$ denotes the Hausdorff measure (volume) of the interval, and $|\partial I|$ denotes the "surface area of the interval", and $\delta$ denotes the diameter of $I$.
I have seen somewhere that $v(I)\leq|\partial I|\times\delta$. Is this true, and how would we prove it?
I can roughly see it in lower dimensions, for instance if $I$ is a rectangle, $v(I)$ its area is certainly less its total perimeter multiplied by the largest distance between points.
Thanks.