A measure $V$ (for "value") is defined on an equilateral triangle. $V$ is absolutely-continuous with respect to the Lebesgue measure, and the value of the entire triangle is $1$.
What is the largest $r$ such that, for every $V$, the triangle contains a rectangle with a value $\geq r$?
One special case is that $V$ is the Lebesgue measure, so the value of a rectangle is its area. The largest area of a rectangle is $1/2$, so $r\leq 1/2$:
However, there is a worse case: if the measure $V$ is spread evenly in a thin strip along the perimeter of the triangle, then any rectangle can contain a value of at most $1/3$, so $r\leq 1/3$:
Is there a measure $V$ for which $r$ is smaller?
P.S. the question is related to both geometry and measure theory. Is there a field that deals with such questions?