Visualizing inequality in arbitrary dimension

Question

In my studies of analysis I was recently stumped by this:

We have a given vector $x\in\mathbb{R}^n$ and I am looking at the inequalities $ 0 < y^T x < 1 $ for all $y \in \mathbb{R}^n $.

I know the inequality $ 0 < y^T x $ means a halfspace separated by a hyperplane $ y^T x = 0 $ but can we visualize $ 0 < y^T x < 1 $? Perhaps there is intuition in $ \mathbb{R}^3 $ visualizing the satisfying region? I thank all helpers.

Answer 1

This reminded me of an old joke

An engineer, a physicist, and a mathematician are discussing how to visualize four dimensions:

Engineer: I never really get it

Phyicist: Oh it's really easy, just imagine three dimensional space over a time- that adds your fourth dimension.

Mathematician: No, it's way easier than that; just imagine $\mathbb{R}^n$ then set n equal to 4.

Perhaps a better visualization is that of a linear function

$$f:\mathbb{R}^n\to\mathbb{R}$$ $$f(x_1,x_2,...,x_n)=\sum\limits_{i=0}^ny_i\cdot x_i$$

Thus the equality $$f(x_1,x_2,...,x_n)=c$$ defines its level sets.

This are the sublevel sets

$$L_{c^-}=\{(x_1,...,x_n)|f(x_1,...,x_n)\le c\}$$ and the superlevel sets $$L_{c^+}=\{(x_1,...,x_n)|f(x_1,...,x_n)\ge c\}$$

Now imagine the intersection of the sets

$$M=L_{0^+}\cap L_{1^-}$$ This is the set of points you are thinking of.