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In theory, such a body doesn't exist. As Urania pointed out, it would have been used long ago. To make a body rotate, you need to either apply acceleration to it or make the body so small (relatively) that it would rotate due to gravitational forces. And, frankly, the question about edges is not relevant here. That's just my opinion.

I wrote that it's possible to translate the problem into a plane and prove that for any point inside a convex polygon, you can draw a perpendicular to one of the sides, so that the end of the perpendicular falls on the side itself, and not its extension.
Well, there you kind of divide this thing into triangles with angles vertex-vertex-center and calculate the heights, maybe it's also worth considering the 360 degrees and so on, I don't feel like solving it further.

Yeah, it's quite possible that it can be done that way. I don't see any significant differences.

Because, as Uranian pointed out, they would have used it a long time ago.

Who knows, maybe people are just stupid and didn't figure it out...

I have a feeling that this is related to the possibility of dissecting a polyhedron into pyramids with vertices at the center of gravity and bases as the faces of the polyhedron (if desired, the faces can be further divided into triangles, thus dissecting the polyhedron into tetrahedra). Perhaps the solid angles of the resulting pyramids or their sum are also involved. But somehow, the idea isn't developing... :(

Yes, one cop figured it out.

The catch is that you need to consider it as a whole. Right now, I'm stuck on the fact that this condition might not apply to a specific faction, but I can't quite figure out how to formulate the condition correctly.

Now, let's try this approach. For a polyhedron to rotate from one face to another, it must have potential energy, meaning the potential energy on some neighboring face must be lower (the center of gravity's position when rotating onto it must become lower). Since the number of faces is finite, the polyhedron can roll from face to face, and its potential energy can decrease. Eventually, it will reach a state where, in the position on all neighboring faces, its potential energy will be greater than in the position on the face it is resting on, and it will no longer be able to roll onto a neighboring face.

Added 2 minutes later
The center of gravity can only descend, and for a polyhedron, it can only descend in discrete steps when rolling, and it cannot descend indefinitely.

Why does the minimum potential energy occur specifically on a face, and not on an edge? While Ment is absent, I'll ask this question for him.:D

Added 2 minutes later
Perhaps I'll answer it myself.

Because the perpendicular from the center of gravity to the edge is longer than the perpendicular from the center of gravity to the face (the hypotenuse is longer than the leg).

Hermit, yes, noted. Sooner or later, the polyhedron will simply minimize potential energy. The physical approach is much simpler here than the mathematical one, through the projection of the center of gravity ;) In this regard, Mort correctly pointed out that it's easier to achieve on the edge. Actually, where I found this problem, they simply stated: "A perpetual motion machine is impossible => this is impossible." But it seemed insufficient to me. What if there's some kind of paradox...? Or maybe this system isn't actually a perpetual motion machine (a hydroelectric power station might also seem like a perpetual motion machine: the river flows endlessly, and we get free energy. But in reality, there's no perpetual motion machine there. Maybe there's some external energy source here too?). I considered potentials to be the optimal explanation. Hermit, make a wish.

Why does the minimum potential energy occur specifically on a face, and not on an edge? Since Ment is absent, I'll ask this question for him.

Added 2 minutes later

Perhaps I'll answer it myself.

Because the perpendicular from the center of gravity to the edge is longer than the perpendicular from the center of gravity to the face (the hypotenuse is longer than the leg).

Well, yes, that's a reasonable addition...

Why not go straight to the top? The main thing is that the potential energy on the adjacent faces should be greater. But in reality, there are no absolute angles in nature, and eventually, it will roll down to a face. I think it's not a necessary condition that it must roll to the face where the energy is lower. I think it's also necessary that the center of gravity is not above the face where it is lying, but what I wrote above clearly proves that it cannot roll indefinitely.

Added 17 minutes later
In a certain kingdom, in a certain state, there lived a king. And he had 100 wise men. One day, the king decided to check how wise his wise men really were. He summoned them to him and said: "Tomorrow, you will be led to the square, lined up, and each of you will be given a red, white, or blue hat, and then the executioner will be called. After that, I will call each of you one by one, and each of you must say what color hat is on your head. Those who guess the wrong color of the hat on their head will have their head cut off!"

The wise men began to think about how to save themselves.

First, they came up with a way to save every other one. For this, he must say the color of the hat of the wise man behind him, and every second wise man must repeat what the one called before him said.

Then they came up with a way to save two out of three. The first wise man looks at the hats of the second and third. If they are the same color, he says that color, and if they are different, he says the color that is not on either the second or the third wise man. The second one comes out and, if he sees that the third one has the same color hat as the first one said, he says that color, and if it's a different color, he says the third color - neither the one the first one said nor the one the third one has. The third one, if the first and second said the same color, says that color, and if they said different colors, he says the remaining color. This way, 66 wise men can be guaranteed to be saved.

Question: what is the maximum number of wise men that can be guaranteed to be saved, and how? Or, if there is no way to guarantee saving more than 66 wise men, prove why!

I would write about your puzzles, but I'm afraid I'll get banned. Hermit, this is a binomial distribution according to Newton. You have to calculate it, it takes a long time, I'm too lazy, and I'm going to sleep.

Don't you feel sorry for the Wizards? :)

Why feel sorry for them? It's all show business.

I have a feeling that the answer is that everyone can be saved except for one, and that one has a chance to be saved. This is based on my memory of seeing the problem, but I don't remember the solution.

3 out of 4 cannot be saved because, according to the first player's answer, 2 of them will not be able to distinguish between "red, red, red" and "blue, red, red". So, unfortunately, 3 out of 2 is the maximum.