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.
- 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. :-)
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.
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:
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.
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.
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.)
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:
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.
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.
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.