Intersection of a straight line and a convex polytope

I apoligize in advance if the question might appear trivial, but there is something unclear to me related to convex polytopes in high dimensions. Let $$P$$ be a convex polytope in $$\mathbb{R}^d$$ for $$d\ge 2$$, and let $$\mathbf{p}$$ an interior point of $$P$$. Let $$L$$ be straight line through $$\mathbf{p}$$.

When $$d=2$$, $$L$$ intersects $$P$$ at one of the its edges (more than one edge only if the intersection point is a vertex of $$P$$).

When $$d=3$$, $$L$$ intersects $$P$$ at one of of its $$2$$-dimensional faces. Hence, the set of all edges (which are ($$d-2$$)-dimensional objects) of $$P$$ does not contain all possible intersection points over the choice of $$L$$ given any $$\mathbf{p}$$ inside $$P$$.

Question: When $$d\ge 4$$, it is always the case that a subset of polygons (which are $$d'$$-dimensional objects for $$d'\le d-2$$) having as vertices the ones of $$P$$ and containing no interior point of $$P$$, contains all possible intersection points of $$L$$?

• There is a line that passes through any two points, so in particular there is a line through $\mathbf{p}$ that passes through any point of $P$. Feb 4, 2021 at 15:43

A ray from an interior point $$p$$ of a $$d$$-dimensional convex polytope will meet the boundary of $$P$$ at exactly one point $$q$$.
With probability $$1$$ the point will $$q$$ be in the interior of one of the $$d-1$$-dimensional facets of $$P$$. But it might be on the boundary of some facet, in which case it will be on several facets.
Any point on the boundary of $$P$$ will be the $$q$$ for some ray since you can always join $$p$$ to $$q$$.