How to detect death in Minecraft in a certain area

depending on the shape of your arena (square or circle), you can restrict a command to run only if entity is within a certain range. i usually use, also, execute as @a at @s, as it quite literally targets (as) everyone (@a) as if (at) they were the individual (@s).

in the instance of a cuboid (square/rectangular area), you'd want to use the x, y, z, dx, dy, and dz arguments (or settings you need to specify) to choose the area you want to limit it to;

• x is the X coordinate of your first corner.
• y is the Y coordinate.
• z is the Z.
• dx is the distance on the X axis to the opposite corner.
• dy is the distance on Y.
• dz is the distance on Z.

so, for an example, if i want to make a player say "hi" while in a 5x5x5 cube area, assuming the center is x0 y3 z0;

execute as @a at @s if entity @s[x=2,y=1,z=2,dx=-4,dy=4,dz=-4] run say hi

this would make it so if someone is in the area between the block at x2 y1 z2 and x-2 y5 z-2, it will run "say hi" as if that person ran the command itself.