# Sequence of nested closed convex sets in $\mathbb R^n$ (a generalization of Cantor´s theorem(axiom))

We have in $$\mathbb R$$ an axiom (sometimes a theorem if we start from another system of axioms) that:

If $$(I_n)_{n=1,2,...}$$ is a sequence of nested closed intervals such that $$l(I_n) \to 0$$ when $$n \to + \infty$$ then their intersection is non-empty (really, the intersection is only one point). Here, $$l$$ is the length of an interval.

However, if the intervals are not closed, then this need not to be true, as an example $$n \to (0, \dfrac {1}{n})$$ shows. Here, the intersection is empty.

There is virtually no reason why we shouldn´t consider sequences of nested sets in $$\mathbb R^n$$, and, most probably, so much is already known about them.

I guess that straightforward generalization of above stated theorem (axiom) also is true, that is:

If $$(C_n)_{n=1,2,..}$$ is a sequence of nested closed convex sets in $$\mathbb R^n$$ such that $$V(C_n) \to 0$$ when $$n \to + \infty$$ then their intersection is non-empty (it should be one point, right?) Here, $$V$$ is the volume (or, would calling it $$n$$-volume be more appropriate?) of a closed convex set.

So, is this true? Is the intersection of nested closed convex sets in $$\mathbb R^n$$ such that $$V(C_n) \to 0$$ when $$n \to + \infty$$ non-empty? Is it one point?

You can start with whatever legal definition of a convex set you want, although, I think that definition:

A set is convex if for every two points $$x,y$$ from that set we have that line segment that connects (joins) those two points is contained in that set.

is the simplest one.

Notify me if this question can be improved, because, it may well be that there are some issues that I am not aware of, and, I am not completely sure that all is right with definitions I gave.

Also, I take as an implicit assumption that every closed convex set in $$\mathbb R^n$$ has an $$n$$-volume.

If this is not true (and I think it is) then suppose that we are dealing only with closed convex sets in $$\mathbb R^n$$ that have an $$n$$-volume.

Also, in our considerations, all closed convex sets are bounded.

Edit: By the comment of Emma, it seems clear that the intersection is only-non empty, it need not be one point, because, if closed convex sets are of dimension $$n-k$$, where $$1\leq k in $$\mathbb R^n$$ then their $$n$$-volume is $$0$$.