The pizza problem is a fairly well-known problem which sounds like this :
You have a circular pizza and you need to cut it such that you and your friend would both receive half of the pizza . Because you're bored you decide not to cut through the center but rather through an arbitrary point on the pizza .You make $n$ cuts through the pizza with angles of $\frac{\pi}{n}$ between them .Each one of you takes one slice alternatively in a clock-wise way. Prove that this slicing is fair for $n \neq 2$ even but for $n$ odd or $n=2$ it isn't .
After seeing the problem I tried it for other pizza shapes also . I observed that the square quickly failed (for two cuts ) and this made me realize that the pizza should be quite 'round'.
Based on this I conjectured the following :
Consider a closed convex set as the 'shape' of the pizza .Also choose $n$ a fixed positive integer .Take an arbitrary point on the pizza and then make $n$ cuts through it with angles $\frac{\pi}{n}$ between them . The friends take the slices alternatively in a clock-wise way. Assume that the friends always get the same amount of pizza for every point on the pizza and any first cut . Prove that the pizza must be a circle .
I realize that maybe this problem (with a convex set) might be too hard so I would be quite happy with any partial results possible (for polygons , triangles , ... I am sure these are a lot simpler )
Thanks for everyone who could help me with this problem . All your help is greatly appreciated .
EDIT I will add some partial results I have found :
$1)$If $n=1$ then it's not possible for any shape .
This should be obvious . Draw a point close enough through the boundary and then a line through it that cuts a small piece of pizza .
$2)$ If $n=2$ and the pizza is a convex polygon with at least one acute angle (or right angle )then it's not possible .
This isn't hard either . Draw a point close enough to the vertex with the acute angle (or right angle) . Draw a cut parallel with one of the sides of the angle and the other cut perpendicular to it . There will be one very big slice one very small slice and two other slices . Now make the point closer and closer to the vertex until the big slice is big enough . This slicing is not fair .