The depth isn't the problem, it's the shape that matters.

A perfectly frictionless curved surface can be treated like a swing or a half-pipe. Increasing the slide distance up to the point of exceeding the top edge just by control of weight and position. It'll take a bit of time to get used to the movement but an agile person should be able to escape a bowl of any size.

What you need to build is something more like a vase, there's a pinch before the rim such that anyone approaching it at speed is merely flung back over the bowl rather than exceeding the rim height.

The king of our country is a modern monarch in outlook and has devised a very similar mode of imprisonment! Deep pits and torture tables and ordinary prisons are much too practical and pedestrian. They also fail to serve their function. The modern despot subscribes to this penal code: It is demonstrated that no criminal punishment may truly serve as neither deterrent nor punishment; for the convict never truly feared our justice nor his punishment, neither the honest man ever needed to be deterred. Therefore, let all punishment of crime be for the instruction and entertainment of the People.

Therefore, the king's mages, taking their cue from the Royal Gardners devise the following means of imprisonment: the Great Globe of Wesparion. Its particulars:

• For the perpetual imprisonment of one to four convicts let there be made a great globe of one palm thick thaumic glass;


• let the globe be made with an equatorial diameter of fifteen feet;


• let there be a neck at the upper pole four hands in diameter at the upper lip, reducing to three hands at the conjunction;


• let there be a broadening at the Great Globe's equator such that it may rest suspended upon a great ring of bronze etched with many powerful runes of doom and woe and which itself shall be suspended from the city's mighty Gibbet of Gilbras the Wise, and which has long served as the last dancing partner of many a justly punished convict;


• let there be a neck at the nether pole two palms in diameter;


• let the royal Guild of Glassblowers make forthwith the Globe and let the College of Wizards inchant the said Globe with their spells and runes of baleful purport and let the said Globe be suspended before our dread Hall of Justice.

The Globe is essentially an overlarge wasp trap, but with the nether chimney facing away from the globe rather than up into it:

Pros:

• Thick thaumic glass is unbreakable through normal means; the frictionless coating prevents the inmates from escaping


• sufficiently wide upper chimney allows (generally naked) prisoners to be slipped inside the Globe; the tapered design prevents them from slipping out again, should anyone try the old Fakir's Rope incantation


• chimney at the top allows small crusts of bread to be dropped in to the inmates, and also allows for occasional showers of fresh water for bathing & drinking


• small chimney at the bottom prevents escape of the inmates, whilst allowing for the escape of noxious fluids & solids


• clear glass, suspended in one of the great squares of the capital city provides hours of slapstick entertainment for young and old alike as the writhing mass of inmates slips, slides, sloshes and slithers about within their inescapable prison!

Cons:

• none.

Note: While in our country, we allow for several inmates to be so housed, I understand your kingdom will allow for only one. I would therefore submit that a sphere of seven or eight foot diameter will be quite sufficient. Once enhoused, none may escape, either by climbing or by jumping.

Kind of like this, only fully spherical, and also unbreakable:

You don't need to use a bowl. If anything a bowl would let the prisoner escape without any external help. A flat circular disk which is taller than the persons with their arms raised would be enough to trap them as long as you dropped them into the center.

The issue with a bowl is you as a person can change your pose, but not your center of gravity (horizontally). Since there is a slight elevation due to the curve of the bowl, a person can stick there arms out and this would cause their legs do slid up the edge of the bowl slightly. Now gravity does its magic. Since the force is no longer directly down, some of it is applied horizontally causing the prisoner to enter an eternal swinging motion. Repeat this enough and they can swing themselves out.

With a Flat disk, the person has no way to create an extra horizontal force. If they stick their arms out, their body will move slightly so that their center of gravity remains in the same place. Once they put their arms down, they will slide back into their original position. If they lie down, their center of gravity will still be in the middle of the disc, so as long as they can't reach the edge of the disc they can't escape.

There are only two issues. Dropping them onto the disk with no horizontal momentum and air resistance..

**A better version of the bowl.

Rather than have the bowl be entirely frictionless, make the base of it have friction. About 1 or 2 feet in diameter so that the prisoner can't jump out of the bowl, and they can't build up enough running speed. This way, if they try to swing out of the bowl, they hit the base and get slowed down. Even better, make it out of sand paper so they really don't want to slid into it.

Just dig a hole. You don't need to mess around with the ideal circumference of a bowl - there's no real benefit. Go out into the garden and dig a six foot hole. Throw the prisoner in. Cast the frictionless spell if you want to be sure he's not climbing out. Put metal bars over the hole to make sure no one pulls him out.

Honestly, the frictionless spell is the least useful part of that setup. It's quite easy to keep a man trapped without it, no spell needed.

How many resources does it take to cast such a spell? I can't imagine that it would be economic considering the slight benefit that a frictionless prison will bring.

Of all the uses a frictionless spell would bring, to use on creating a prison seems like one of the least.

Make detainees be their own prison

Why bothering to construct a prison at all?
Let me tell you the story of Tom. Tom was found guilty of having bad culinary taste, which is an non pardonable crime in the kingdom.

Soon after his trial, Tom was taken to the market place of the city. There, mages enchanted Tom's skin, so that it became friction-less. And then the mages walked away.

The crowd, that had hope for a good old execution, was not very impressed at first. They whispered: well, can't he just escape now?. But they soon understood what Tom fate had become. Tom tried to walk but his feet couldn't push on the floor. He lost balance and fell, and just kept on slipping when he tried to get up.

Tom raged and cried, but he wouldn't give up. Minutes passed and the crowd was staring uncomfortably at him. He started to beg passers for help. Most would look away, but some did try to get Tom on his feet, but that was of no use: Tom would grab their hand, and that hand would slip away too.

And that's when the crowd started to truly fear magic. Rumors spread that Tom could be contagious, and soon no one would approach him.
The few who had him in pity would bring food every now and then, but would have to place it directly in his mouth, for he wouldn't be able to grab anything.

Sure he could move about by sliding on the floor effortlessly. But Tom could not work. Tom could not have felt a hug, if anyone had been kind enough to give him one. So he went to the nearest cliff and he let himself slide one last time. He did not feel the air brushing past him as he fell. No more did he feel the water he fell into, but the ocean did close his tomb and the water did rush into his lungs. And then he was truly gone.

And the city resolved never to talk about Tom again, for it was a painful memory that no one wanted to face. This punishment was abolished and instead the spell started to be used for the common good of the kingdom instead. Prosperity went up, crimes went down and soon the budget allocated to prisons was not a major problem.

Others have spoken on the geometry (and other things) but I would go with a frictionless field or area, rather then a surface.

One big issue is that such a spell would have far ranging applications, both in military and civilian use. Some examples:

• Catapult parts that reduce friction


• frictionless skis to move heavy things


• conveyor or slides, to transport things


• arrow heads (shot or other missiles)


• armor plates


• pots and pans ( lol )


• moats and defenses (frictionless walls)

etc.

By affecting an area instead of a surface, it actually makes a better prison and gives less utility to the spell. Anything within this area is frictionless.

So throw the guy a rope, well he cant hold it, can't tie it etc. It may make eating and doing other biological activities a challenge, but so what. Prisons aren't meant to be comfortable.

However a field that makes everything frictionless within it would give it some major drawbacks in other uses. Some of these could be gotten around with multiple small fields instead of one large field but it would probably require great skill on the part of the mage.

Take the ski idea:
A large field makes all the stuff on the sleigh slide off. But you could do dozens of small fields only on the ski's themselves. But that would take more time casting, more precision, and presumably more skill on the part of the caster.

The point is frictionless tech (or magic) would revolutionize the world, even in stories things (technologies) should not exist in vacuum. People are resourceful they will find other uses for it. For me at least, when i see something like that in a story and it has no other impact on society, it really jumps out as something that was done for "plot reasons".

I'm a physicist, so I'm going to give a physicist's perspective, considering a few factors. First, I'll answer the question. Next, I'll discuss the physics of a frictionless surface.

Diameter of Bowl / Shape of Depression

The diameter of the bowl does not matter at all, nor does the shape, provided it is wider everywhere than the person is tall.

My suggestion for elegant simplicity is a half-sphere bowl.

Slope of Walls

The walls must be sloped enough or steep enough such that there are no true handholds or footholds (i.e. ledges/steps), because then the lack of friction would not matter, as they could pull or push themselves upward.

In fact, you could even have a depression with vertical walls (not sloped).!Sloped walls make it harder to escape by jumping or climbing out, but they also allow for a strategy that is described in the Examples, below.

Depth of Bowl

The depth of the bowl (surface to lowest point) is critical.

For one prisoner: build the bowls a little higher than the prisoner's fingers when they reach as high as they can while standing.

Other Considerations

• The prisoners should not have anyone with them.
  
  
  
  
  • If they have a single partner, make the bowl just higher than either of them can reach while jumping as high as they can.
  
  
  • If they have more than one person in the bowl with them, make sure the height of the bowl is approximately $pi approx 3.14159...$ times the average height of the prisoners (or the tallest prisoner height, to be safe). If the prisoners have very different weights, multiply this height by the ratio of the heaviest prisoner's weight to the lightest prisoner's weight.
  
  
  
  


• Make sure the area around the bowl does not have anything which a rope could attach to.


• Do not let prisoners have many things with them in the bowl.


• Make sure the prisoners have no way of convincing outsiders to help. :-)

Upkeep

As long as the frictionless surface is maintained, these bowls would require no upkeep save sanitation (urination, excretion) and/or feeding, as others have said.

One easy way to allow for sanitation is to put a drain in the depression and make the depression always slope downward (even the slightest bit!) towards the drain. For privacy, one could put a little hut around the drain (if so, the bowl would need to be deeper because the prisoner could gain horizontal momentum by pushing off it) — never mind how the prisoner will actually get inside the hut.

Feeding could be done by throwing or dropping food towards the prisoner. A clever prisoner could throw the food and gain some horizontal momentum, meaning the depression would need to be a few inches deeper.

Impressiveness

My mental image is something like 20' x 20' with a depth of 9' and walls with a radius of curvature equal to the depth. The large-ish cross section gives the prisoner room to play with the idea of getting out, and their attempts would be amusing and entertaining to those in power (or their guests).

A different design would be a half-sphere 9' in depth.

The impressiveness in this design is how simple the design is and how weak and helpless it makes those inside look.

Discussion / Rationale

The thing preventing people from getting out, of course, is that if they somehow were to move away from the bottom of the bowl, where gravity keeps them trapped, the curved walls would mean that they would lose momentum trying to get out because gravity would act against their motion.

Because there is no friction on the bowl's surfaces for them to walk, the only way for them to move on their own, without outside help, is for them to push against something else really hard -- an "inelastic collision" in physics language.

Here are two examples of what this could look like:

Example A

Suppose there were two prisoners, contrary to the premise you described. They could push against each other really hard to fling themselves backward. If they did so with enough force, they could slide up the side of the wall until gravity brought them back down again with a crash.

In Example A, the acceleration each prisoner receives as a result of pushing off each other depends on their own strength and that of their partner, as well as their own mass. A strong, heavy person would be able to launch a weak, scrawny partner further than the weak, scrawny partner would be able to push the strong, heavy person.

Because legs generally have the strongest muscles in the body, they can get more acceleration by pushing off a partner (or a wall in the middle of the bowl??) with their legs than they'll ever get by throwing something with their arms. Their force of pushing off of something with their legs would be approximately equal to the force they use to push off the ground when they jump as high as they can. In fact, because no energy is lost to friction, the maximum height they could reach on the walls of the bowl would be equal to their max jump height. The bowl depth would need to be only a few feet (~1 meter) more than how high the prisoner's hands could reach during their highest jump — about 7–9 feet for most humans.

Example B

A single prisoner has a heavy item with them. They throw the item forward as hard as they can. In doing so, they push themselves backwards. As in Example A, if they do so with enough force, they can go up the sides of the wall.
If there was only one prisoner, then throwing something would buy them no more than a foot of height up the sides of the bowl — unless they had superhuman throwing strength.

Example C

If there's one prisoner, the prisoner can take off their clothing, rip it into long, thin strands and tie the strands together, forming a makeshift rope. They could then throw the "rope" out of the bowl, and with much luck (or skill), they manage to secure that end of the "rope" to something immoveable outside the bowl. They then pull on the "rope" to get out of the bowl. (They would not be able to "climb" out of the bowl, because their feet would have no friction with the bowl, hence they'd have to pull themselves out.)

Example D

Suppose there are many prisoners in a given bowl. They could arrange themselves such that they formed one secure line, where prisoners in the middle of the line would have their feet held on one end and would hold somebody else's feet above their head. Assuming they were all relatively strong (such that the structure did not break or buckle under their combined weight), they would be able to reach the surface when one person's hands reached the top.

Assuming they all have similar weights, the line of humans would have its middle at the bottom of the bowl. This means that the bowl height would have to be just over $pi approx 3.14159...$ times the average height of the prisoners to keep them in.

If they had different weights, they could arrange themselves from heaviest to lightest for an advantage. Gravity would pull the heavier prisoners on one end of the line closer to the lowest point in the bowl, which would then move the lighter side of the line closer to the top. In this case, you'd need a deeper bowl, where the new height is multiplied by the ratio of the heaviest prisoner's weight to the lightest prisoner's weight (or thereabouts).

Physics of a Frictionless Bowl

Some answers, comments and chat have discussed the possibility of the prisoner "building up momentum". Obviously, being able to have sufficient momentum along the surface of the bowl would allow the prisoner to escape.

Every force that a person exerts on a surface can be decomposed into a parallel and a perpendicular force:

Perpendicular Forces

If the prisoner pushes against the surface either by gravity or because they're using their legs to jump off the surface, then the surface exerts an equal-and-opposite force (a normal force) on the prisoner. If the prisoner pushes hard enough, the normal force will be enough to launch them perpendicularly away from the surface. In a bowl-shaped depression, the walls nearest to the rim are nearly vertical. If a person jumped perpendicularly off a nearly vertical surface, they would just fly nearly horizontally back towards the center of the bowl.

Parallel Forces

To build up momentum along the surface, the prisoner would need to experience an external force that is parallel to the surface. Ignoring any outside helpers, and ignoring air resistance (which is absolutely negligible for this sort of problem), the only two forces the person would experience are friction and gravity. Because friction acts parallel to the surface, and because the surface is frictionless, the only external force that could possibly act on the prisoner along the surface is gravity.

Gravity

The prisoner can only build up momentum via an external parallel force, and because friction is only parallel, gravity is the only (sometimes-) external force that has a component parallel to the surface that could help the prisoner get out of the bowl.

Gravity is a conservative force, which means that no matter what path the prisoner takes to get from one height to another height — whether by jumping straight up from the center of the bowl or by sliding upward along the curved wall — the change in energy (and therefore the change in vertical momentum) is the same for all paths. A corollary of this is that the sum of their kinetic energy (related to momentum) and their gravitational potential energy is constant unless an external force acts on them. For this reason, the highest a person can move their center of mass from the bottom of the bowl is equal to however high they can jump plus some for any horizontal motion they started with.

Skaters and Half-Pipes

In the case of a skater "building up momentum" in a half-pipe, the skater pushes directly downward so that the surface pushes them back upward. This reactionary force/push from the surface has a component parallel to the surface (mostly upward), due to friction, and a component perpendicular to the surface (mostly towards the center of the half-pipe), called a normal force. The parallel component (friction) pushes them upward and lets them build momentum to get higher and higher, while the normal component pushes them off the surface slightly.

In the case of a frictionless half-pipe — like our bowl-shaped prison — there is no friction, so the person can only push themselves perpendicular to the surface, and will never be able to build up momentum.

What I pictured when I read this question was something like a tea saucer with a person in the middle, set out in public for the amusement of a crowd. Perhaps you could even have rows of them like pillars in a particularly intimidating waiting room at the justice department.

It should be noted that the appeal of this system is the lack of any barrier at all between the criminal and the crowd; in fact the criminal stands at eye level, not down in a pit, visible from across the square and not even easily distinguishable from the crowd until you see what he is standing on.

This already introduces some fundamental problems: in addition to the swinging escape method discussed elsewhere, any member of the crowd could pull the man out or fling him a useful tool at any time. Any intercourse with the public will necessitate guards. Even elemtilas' globe solution could probably be broken open if left totally unprotected in a public place overnight.

Guards would also be needed to give the prisoners food at mealtimes, so we can assume some external help for disposing of their... other needs. Perhaps prisoners could be provided a small cup for hygenical purposes; flinging it should not give too much momentum. At least on a frictionless surface cleanup should be easy.

But to the question: could the dishes be safely stored in rows in a dungeon warehouse without guard interaction overnight? What geometry does the dish need to have to make swinging out impractical?

The simplest solution is to put something in the center to dampen momentum. This could be a puddle of water or even a small bump (like a 'punt' in a wine bottle), so long as anyone trying to swing hits it on each swing and is slowed down appreciably, and they can't use it to push off - at least not all the way to the edge. In this system, if we successfully prevent any type of swinging from starting, the edge only needs to be safely out of reach of someone in the center. Exactly how far someone might be able to 'push off' depends on the exact nature of our momentum damper in the center, but I can't imagine needing a radius of more than 10 feet if we've prevented swinging, and even that's only if someone somehow managed to brace their feet securely against the center, which ought to be impossible.

As for the necessary depth of the bowl or dish: to get out, a person essentially needs to be able to lift their whole weight this distance. It's easy to build up that amount of work incrementally through swinging, but to get it all in one go by pushing off our momentum damper would be difficult. You could push against a frictionless 'punt' at the proper angle and get some momentum, but how much? In the best case, you would just sit on top of it and slide down, so as long as the punt is shorter than the rim, all should be well, even if the rim is only a few inches above the lowest point in the saucer.

To continue your experiments, I'd start with a frictionless saucer, 20 feet in diameter, 1 foot high (from the lowest point to the outer rim), with a 2 foot diameter punt in the center, 6 inches high. Very chic.

Note that trying to move one of these things while occupied will result in a lot of sliding around, and perhaps out; consider temporary bars for excursions.

Whoever wants to pull a rope to help some prisoner to escape has to rely on friction. Without friction they cannot pull. So one could think of a frictionless area around the confinement zone. But that could be easily circumventend by just moving out of it.

Even assuming this could be taken care of by design, they could still use reaction to exert a force on the prisoners while being within the frictionless area, so something like a jet pack wouldn't care about lack of friction.

Therefore I would say that the most effective, practical and cost effective design is simply a well with frictionless vertical sides and bottom, and the well top covered by a grid large enough to let slide down a bowl of soup and nothing more.

In my opinion no matter how big the bowl is or what shape it is, it would still be possible to just throw a rope down to the prisoner and just pull them up. This would be quite easy for the rescuer because there is no friction (obviously), so having an open-air bowl wouldn't be the ultimate prison. But there are ways to make it harder for the rescuer.

1. The lip. The rescuers biggest enemy will most probably be gravity, because it will be fighting against their strength and slowing down the rescue. To take advantage of this, the bowl should integrate an inwards facing lip that forces the rescuer to lift the prisoner vertically in midair at some point.

But what if the rescuer is super buff?

2. Make the prisoner heavy. Force the prisoner to wear a straitjacket with weights, making them impossible to lift by a person. This could also be handy for they couldn't grab onto a rope, forcing the rescuer to lasso or grapple onto the prisoner, possibly hurting them.

But say the rescuer successfully grabs onto the prisoner and has buddies to help him lift up the prisoner.

3. Restrict the area around the bowl. This means that the rescuer can't have a group of people helping them or a large machine parked next to the bowl. This also means that if the rescuer decides to have lots of people there or something, the chance of someone falling into the bowl because of the restricted area could be higher.

The shape is important. Make the prison a sphere, with the entrance at the top and a small vertical corridor of same material before the door.

Or just add a little oil.

You should build it to be a bowl with a pillar in the middle. The pillar addresses the other answers' concerns about building up momentum, provided it is neither tall enough nor wide enough to provide an escape by itself. This also allows your prisoners some dignity, by letting them have a place to sit that isn't frictionless, in case you care about that. It can also serve as their privy.

This also has other benefits: it's easy to remove a prisoner that is to be released dies because entering the bowl is easy with a rope - the side of the bowl needn't be too steep. Consider that to move upward 3 feet is pretty much impossible for any normal person, and the fact that it's at a shallow angle doesn't change that. So if you made wide, shallow bowls with frictionless sides maybe 12 feet radius, 6 feet deep (a 6' person would effectively have to jump 3' vertically to get any purchase), you'd have essentially escape-free prisons.