Given a bounded measurable subset of the plane—the pizza, find two perpendicular lines that divide it into four equal areas.
This is always possible.
Consider a line L, initially horizontal.
Of all the lines parallel to L, one will bisect the pizza.
Likewise we can find another line M, perpendicular to L that also bisects the pizza.
We are not done for the pizza piece P that is to the left of both lines may not have an area of ¼.
That piece will have the same area as the opposite piece.
Consider the piece Q to the left of L and to the right of M.
The sum of those two areas is ½ since L bisects the pizza.
Rotate L continuously and consider the intermediate constructions.
The areas P and Q will vary continuously (since the pizza is bounded) and these area magnitudes will have interchanged when L becomes vertical.
By continuity they must have been equal for some choice of L.
**Q.E.D.**

We can divide the same subset into 6 equal pieces with three concurrent lines.
Choose L horizontal again.
Choose M to bisect the pizza such that the piece to the left of L and the right of M has area 1/6.
Choose a third bisecting line N such that the piece to the left of L and the right of M has area ⅓.
We are not done for the lines may not be concurrent.
Again rotate L until it is parallel to where M started, and watch the triangle formed by the three lines.
The new triangle will be the same triangle but L will be a different side of that triangle, which will have moved to the other side of L.
In that continuous process the area of the triangle will have gone thru 0, which provides the solution.
**Q.E.D.**

I think a pizza can be divided into 7 equal pieces by three non concurrent lines. Start with a family of lines that cut off 3/7 of the pizza.

There are not enough degrees of freedom to generally divide a 3D blob into 8 pieces by perpendicular planes. I think that a regular tetrahedron cannot be so divided.

You have a square table with 4 long legs. You have an island which is ‘big enough’. There is a place on the island where you can place the table so that it is level and each leg touches the ground.