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Totoro
Totoro
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Let $(M^n,g)$ be a closed smooth Riemannian manifold. Consider a set $\mathcal B_{\epsilon}$ which isconsists of a maximal collectionnumber of points in $M$ with pairwise distance no smaller than $\epsilon$.

We define the measure $$ m_{\epsilon}:=\frac{\sum_{x\in \mathcal B_{\epsilon}} \delta_{x}}{|B_{\epsilon}|}. $$ Can we prove that as $\epsilon \to 0$, $m_\epsilon$ converges weakly to the $n$-dimensional Hausdorff measure up to a rescaling?

Let $(M^n,g)$ be a closed smooth Riemannian manifold. Consider a set $\mathcal B_{\epsilon}$ which is a maximal collection of points in $M$ with pairwise distance no smaller than $\epsilon$.

We define the measure $$ m_{\epsilon}:=\frac{\sum_{x\in \mathcal B_{\epsilon}} \delta_{x}}{|B_{\epsilon}|}. $$ Can we prove that as $\epsilon \to 0$, $m_\epsilon$ converges weakly to the $n$-dimensional Hausdorff measure up to a rescaling?

Let $(M^n,g)$ be a closed smooth Riemannian manifold. Consider a set $\mathcal B_{\epsilon}$ which consists of a maximal number of points in $M$ with pairwise distance no smaller than $\epsilon$.

We define the measure $$ m_{\epsilon}:=\frac{\sum_{x\in \mathcal B_{\epsilon}} \delta_{x}}{|B_{\epsilon}|}. $$ Can we prove that as $\epsilon \to 0$, $m_\epsilon$ converges weakly to the $n$-dimensional Hausdorff measure up to a rescaling?

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Totoro
Totoro
  • 2.5k
  • 9
  • 19

Average of the sum of dirac measures

Let $(M^n,g)$ be a closed smooth Riemannian manifold. Consider a set $\mathcal B_{\epsilon}$ which is a maximal collection of points in $M$ with pairwise distance no smaller than $\epsilon$.

We define the measure $$ m_{\epsilon}:=\frac{\sum_{x\in \mathcal B_{\epsilon}} \delta_{x}}{|B_{\epsilon}|}. $$ Can we prove that as $\epsilon \to 0$, $m_\epsilon$ converges weakly to the $n$-dimensional Hausdorff measure up to a rescaling?