I Intersection of a 4D line and a 3D polyhedron in 4D

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The intersection of a 4D line segment and a 3D polyhedron in 4D is indeed a point if they intersect, assuming they are not confined to the same 3D space. This conclusion aligns with the concept that a line in a higher dimension can intersect a lower-dimensional object at a single point. To prove this, one can analyze the situation using a coordinate system that aligns the 3D polyhedron with the axes. The reasoning parallels that of a line intersecting a plane in 3D, extended to an additional dimension. Thus, the intersection remains a singular point in 4D space.
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Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
 
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If they are not in the same 3D space then the intersection will be a point. You can even use the full 3 D volume and a full line and it will still be a single point.

You can look at the line segment in a coordinate system where the 3D object is aligned with three axes, for example.
 
Thanks. Is there a mathematical proof for your observation? A hint would also work.
 
My second paragraph is a description how to get to a proof.

It's basically the same as with a plane and a line in 3D, just with one more coordinate.
 
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