I'm having some trouble picturing what some sets (such as lines/planes or arbitrary sets) look like when projected onto a line or a plane.
The particular example I have at hand is from a paper of Imre Barany regarding intersections of pairwise intersecting convex bodies in $\mathbb{R} ^3$.
Given a collection $F$ of pairwise intersecting convex sets in $\mathbb{R}^3$. Projecting the sets of $F$ onto a line gives a collection of pairwise intersecting intervals. These intervals have a common point. let that point be $p$. The plane whose projection onto the line is $p$ intersects all the sets of $F$.
I can understand how projecting convex sets onto a line gives a collection of intervals (using convexity), and why these intervals have a common point. What I'm having trouble seeing is why there is a plane whose projection onto the line is $p$, or more generally what does a projection of a plane onto a line look like? a point? an interval?
Does this have anything to do with the dimensions of these sets? For instance, a plane is a 2 dimensional set in $\mathbb{R}^3$ and a line is 1 dimensional. I would expect their intersection to generally be a point. And for 2 affine sets of dimension $d-k$ and $d-m$ in $\mathbb{R}^d$ I would expect the intersection (again, it might not exist and these sets might not be in general position but assuming they are) to be of dimension $d-(k+m)$ since the intersection has to satisfy both the k equations of the first set and the m equations of the second set.
I don't know if there is a way to imply a similar "rule" for projecting a $k$ dimensional affine set onto an $m$ dimensional affine set.
I would appreciate an explanation and any other insights that could help with this.