Recently I baked a spherical cake (3cm radius) and invited over a few friends, 6 of them, for dinner. When done with main course, I thought of serving this spherical cake and to avoid uninvited disagreements over the size of the shares, I took my egg slicer with equally spaced wedges(and designed to cut 6 slices at a go; my slicer has 5 wedges spaced at 1 cm apart) and placed my spherical cake right in the exact middle of it before pressing uniformly upon the slicer. Was I successful in avoiding the feuds over the shares. If so then what could be the mathematical explanation of it and if not then why not? My Setup looked a bit similar like given in the image below:
I assume you mean something like this (although this is intended for five persons). If so, then that is obviously not a fair way of dividing the cake (the other slices will be much smaller than the inner slices).