Alice and George divide a cake between them. The cake is a 1-dimensional interval and both players value the entire cake as 1. The valuations of the players are represented by non-atomic measures on the cake.
The following division procedure is used. The procedure takes a parameter $v\in [0,1]$:
- Alice marks a piece at the left of the cake, which is worth exactly $v$ for her. George does the same.
- The leftmost mark is selected, breaking ties arbitrarily. Suppose w.l.o.g. that the leftmost mark is Alice's. Then the cake is cut at Alice's mark and Alice receives the piece to the left of the cut.
- If the piece to the right of the cut is worth at least $v$ for George, then George receives this piece and the division is considered a "success".
- Otherwise, the division is considered a "failure", since it is not possible to give both players a value of $v$.
Note that the leftmost piece is worth at most $v$ for George, so the rightmost piece is worth at least $1-v$ for George. Hence, the procedure always succeeds for every $v\leq 0.5$.
Let $V$ be the set of all values $v$ for which the procedure succeeds (for fixed valuations of Alice and George). Obviously, $V$ is an interval, and $[0,0.5]\subseteq V\subseteq [0,1]$.
My question is: Is $V$ always a closed interval?
(Equivalently: is there a maximum value for which the division procedure succeeds?)