Ham sandwich theorem for other ratios than 1:1? The Ham Sandwich theorem Let $A_1,\dots,A_n$ be compact sets in $\mathbb{R}^n$. Then there exists a hyperplane that bisects each $A_i$ simultaneusly into two pieces of equal area measure.
A possible generalisation comes to mind:
a) Given $0<r<1$ can we in general find a hyperplane that divides each $A_i$ into two pieces with the given ratio between them.
This should be false (it is an exercise in "Using the Borsuk-Ulam lemma" by Matousek). However I haven't been able to come up with a counterexample. How would I show this is false for all $r\ne 1/2$?
We may also ask about further generalisations:
b) What about specifying ratios $r_1,\dots, r_n$ for each set?
A counterexample here would be take all the $A_i$ to be the same set. What if we add the requirement that the $A_i$'s are disjoint and "far apart" from each other, say with distance larger than the sets diameter? In this case it seems we should at least be able to specify one of the ratios to to be arbitrary.
Consider as an example the the ratios $(r_1,r_2) =(1/3,1/2)$ in $\mathbb{R}^2$. To get the first ratio we add a compact set $B$ of area $A_1/3$ to $A$, creating a new set $A'_1 = A_1 \cup B$. By placing $B$ sufficiently far away in a direction such that a line can't intersect $A_1,A_2,B$ simultaneusly, the 1:1 bisection of $A'_1$ should give the desired split of $A_1$. Is this reasoning correct?
 A: Your reasoning is correct!
However, your condition with the distances can be relaxed: the only reason you need this in you proof is that you want a place to put the additional set that it is not intersected by any line passing through the two other sets. This is possible whenever the two original sets can be separated by a line. In this case your reasoning can be generalized to arbitrary ratios for both sets.
This is a special case of a „generalized Ham sandwich theorem“ by Bárány, Hubard and Jerónimo (Paper), which also holds for point sets, as shown by Steiger and Zhao (Paper). The problem is also known as the $\alpha$-Ham sandwich problem in the literature, e.g. here.
In the plane, if you want the ratio for $A_1$ to be 1/2, then you can also say the following: if there is a line bisecting $A_1$ which has an $\alpha$-fraction of $A_2$ ($\alpha\leq 1/2$) on one side, then all ratios in $[ \alpha, 1-\alpha ]$ are possible for $A_2$. This follows from the intermediate value theorem.
Finally, as for the counterexample to arbitrary fractions without some conditions of separatedness, you can for example distribute the first mass evenly in some annulus and let the second mass be almost a point at the center of the annulus. Now, every line passing through the second mass will cut the first one roughly in half.
