I'm not sure how best to phrase my question. I'll explain first what I mean by the title.
If we work on an interval $(a,b)$, then by choosing a number uniformly on this interval we divide the length "uniformly" - that is, the area of each part will be uniformly distributed on $(0,b-a)$. I'm not sure if there is a better word for what I mean.
I am interested in doing the same for general convex sets in $\mathbb{R}^n$, though would also be interested in any partial results e.g. just for cubes/hypercubes, low dimensions bigger than $1$, etc. Is there a well known algorithm for choosing a random hyperplane such that for a general convex set $C$ with area $A$, which will be subdivided into sets $C_1,C_2$ with area $A_1,A_2$, that we can have $A_1 \sim U(0,A)$?
Thanks for your help, results partial or otherwise are appreciated - the speed of the algorithm isn't my main concern, I just want to know what (if anything) is out there as I'm struggling to find things at the moment.
Given a convex set and a line in some direction, consider the set of parallel hyperplanes normal to that line. The area of intersection with the set is a continuous function of the intersection of the plane and the line. You can use that function to define a probability distribution on the interval $[0,A]$ choosing from which gives you the distribution you want.
So there are many such distributions - one for each direction.