I am trying to verify that the density of the cone in three space has 2 density $\sqrt{2}$ at the origin.
The m-density for a set $A\subseteq \mathbb{R}^n$, with $1 \leq m\leq n$ is defined $$ \Theta^m(A,a)=\lim_{r\rightarrow 0}\frac{\mathscr{H}^m(A\cap B^n(a,r))} {\alpha_mr^m} $$ for $\alpha_m$ the m volume of the unit ball.
So for the cone, we have $$ \Theta^2(C,0)=\lim_{r\rightarrow 0}\frac{\mathscr{H}^2(C\cap B^3(0,r))} {\alpha_2r^2}=\lim_{r\rightarrow 0}\frac{\mathscr{H}^2(C\cap B^3(0,r))} {\pi r^2} $$ I am unsure how to evaluate the numerator, it is given by $$ \mathscr{H}^2(C\cap B^3(0,r))=\lim_{\delta \rightarrow 0}\inf_{C\cap B^3(0,r)\subseteq S_j, diam(S_j)\leq \delta} \pi \sum (\frac{\text{diam}(S_j)}{2})^2 $$ Where the infimum is taken over all possible coverings with each $S_j$ of diameter smaller than $\delta$.
It would seem that the diameter of any cover could get arbitrarily small, since the cone collapses to a point at the origin and therefore the intersection with the unit ball of radius $r$ again ought to be just a point. Is this intuition wrong? How do I compute the measure?
edit: I picked this point since it seemed to be easier, please correct me if I am wrong. I would also be happy for guidance in showing that the density everywhere else on the cone is 1.
You could use area or co-area formulas.
In practice one knows that the $H^2$ Hausdorff measure coincides with the usual area for sufficiently regular sets.
https://en.wikipedia.org/wiki/Coarea_formula