There is a cake with $n$ toppings. I want to cut a small piece out of each topping, such that the remaining cake is connected. Is this always possible?
SOME FORMAL DETAILS (possibly not all of them are relevant for a general solution):
- The cake is a square.
- All toppings are pairwise-interior-disjoint axis-parallel rectangles.
- The pieces should be pairwise-interior-disjoint axis-parallel rectangles.
- The size of each piece should be more than 0 and less than half of the size of the corresponding topping.
1-DIMENSIONAL CASE:
when the cake, toppings and pieces are all 1-dimensional segments, the answer is NO. Specifically, assume the cake is the segment $[0,n]$, and the toppings are the segments $[i,i+1]$ for $i \in \{0..n-1\}$. It is clear that any attempt to cut a small piece from more than 2 toppings will create a disconnected remainder. My question is: is this result also valid in 2 dimensions?
The problem seems to trivial. Let $A_1,\dots, A_n\subset [0;1]^2$ be the toppings such that $\operatorname{int} A_i\not=\emptyset$ for every $i$. We can easily choose points $z_i\in\operatorname{int} A_i$ for every $i$ such that no two different points $z_i$ has equal abscissas. Therefore we can slightly inflate each point $z_i$ to a small rectangle $R_i\subset \operatorname{int}A_i$ having mutually non-intersecting orthogonal projection on the horizonal side of the square. Then the boundary of the square $[0;1]^2$ is contained in the set $L=[0;1]^2\setminus\bigcup_{i=1}^n R_i$ and each point from $L$ can be connected by a vertical line with the boundary.