Level sets of higher N

Question

This may be a dumb question and it's essentially nomenclature curiosity but everywhere I see a level set defined, it's what we all know.

For a function f:Rn->R, its level set for a value k is the set: {(x1, x2, ..., xn): f(x1,x2,...,xn)=k}

if n = 2, this is a level curve.

if n=3, this is a level surface.

However, I've never seen any place describe what we would call if n>3.

How would we call this for n=4 and n=5?

Answer 1

Hi John S.,

It's very simple: we don't have a good way of conceptualizing an n>3 level set. It would be visualizable just fine as a 3-D construct embedded in a 4-D space, but we can't wrap our puny brains around that. You can try to step rationally up the respective spaces, with ideas such as "if I cut this object with a simpler figure, what will the intersection look like?" But that's different than being able to conceptualize it as a whole, isn't it.

-- Cheers, --Mr. d.