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. 



Ham sandwich theorem for discrete measures - reference request
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.


