Ham sandwich theorem for discrete measures - reference request

Question

A discrete version of the ham sandwich theorem states as follows (see for instance "Common Hyperplane Medians for Random Vectors" - Hill):

For every $\mu_1,...,\mu_n$ discrete (i.e., purely atomic) probability measures on $\mathbb{R}^n$, there is a hyperplane $H$ defined by $\sum_{i=1}^n a_i x_i =b$ such that for every $1 \leq j \leq n$, we have that $$ \mu_j (\lbrace (x_1,...,x_n ) : \sum_{i=1}^n a_i x_i \geq b \rbrace ) \geq \dfrac{1}{2} \text{ and } \mu_j (\lbrace (x_1,...,x_n ) : \sum_{i=1}^n a_i x_i \leq b \rbrace ) \geq \dfrac{1}{2} .$$

My question is about the error when one considers open half planes. I am looking for a result of the following nature:

Let $\mu_1,...,\mu_n$ are discrete probability measures that are supported on finite sets (I don't know if the assumption on the support makes a difference). Assume there is $0 <\delta$ such that for every $x \in \mathbb{R}^n$ and for every $j$ we have $\mu_j (x) \leq \delta$, then there is $\varepsilon = \varepsilon (\delta)$ and a hyperplane $\sum_{i=1}^n a_i x_i =b$ such that for every $j$, $$ \mu_j (\lbrace (x_1,...,x_n ) : \sum_{i=1}^n a_i x_i > b \rbrace ) \geq \dfrac{1}{2} - \varepsilon \text{ and } \mu_j (\lbrace (x_1,...,x_n ) : \sum_{i=1}^n a_i x_i < b \rbrace ) \geq \dfrac{1}{2} - \varepsilon ,$$ and $\lim_{\delta \rightarrow 0} \varepsilon (\delta ) = 0$.

I assume that this should be a known result, but I haven't found a reference yet.

Answer 1

You could not find a reference because the statement is not true. Suppose that we are in two dimensions but both are measures are concentrated on a line. The first measure is uniform on $1,\ldots,k$ while the second measure is uniform on $-1,\ldots,-k$. In this case any open halfplane has measure $0$ or $1$ with respect to one of the metrics.

If you add the condition that the points in the support are in a general position, then the statement follows trivially from the continuous version.