What is a hypercylinder?

Question

Wikipedia says, A hypercube is an n-dimensional analog of a square/cube. What is a hypercylinder then? An n-dimensional analog of a cylinder? Constant Approximation Algorithm for MST in Resource Constrained Wireless Sensor Networks gives a definition,

The Isolation Property. Let c> 0 be a constant. Let E be a set of edges in k−dimensional space, and let e ∈ E be an edge of length l. If it is possible to place a hypercylinder B of radius and height c .l each, such that the axis of B is a subedge of e and B ∩ (E − e)= φ, then e is said to be isolated. If all the edges in E are isolated, then E is said to satisfy the isolation property.

Does it mean, if I have edge sets E and then take a subset e out E, then if I'm able to place, a some sorts of martian object, B in the so called k space along with an element of e, then B cannot contain any elements from E-e ?!

Thanks in advance

Answer 1

First, in n dimensional geometry, it is common to define an object using it's features in 2 or 3 dimensions (i.e., by writing an n dimensional definition, which,when restricted to the case of $n=2$ or $n=3$ is constant with the 2 and 3 dimensional objects we know). In the case of a cylinder, this object, in 3D is a circular disk or radius $r$, lying on a plane, that has been extended by a distance, $h$, along a vertical, perpendicular axis. In n dimensions, we proceed as follows.

Take an n dimensional plane $a_1x_1 +a_2x_2+...+a_nx_n=c$ (where the x's are variables and all the other letters are constants). Choose a point,$t$ on this plane to be the centre of the base. The base will be all those points on the plane whose distance from $t$ is less than or equal to $r$ (where distance is measured using the Euclidean metric). From linear algebra, we can construct a vector, $v$ which is perpendicular to this plane. I will refer to the "top" side of the plane as the side in the direction of $v$. The cylinder is all those $x\in{R^n}$ that are "above" the plane, whose projection onto the plane, along $v$ is a point on the base, and the minimum distance from $x$ to a point on the base is less than or equal to $h$.

For you second question, the "Isolation Property" seems like a topological definition. It sounds very similar to the following: "For every $e\in E$ there exists an open neighborhood of $e$ not intersecting any other element of $E$.

This is an n dimensional way of saying that each edge has a little bit of space around it on all sides (while this definition is clear in 3 dimensions, you need a rigorous description in higher dimensions).

Does that help?

Answer 2

But can't a hypercylinder also be an $n$-dimensional objects with more than one linear dimension? By Joe's definition, a hypercylinder just increases the dimensions of the circular subspace, but leaves the linear one to $1$.

I'm not sure about the concrete definition of a general $n$-dimensional cylinder, the one we know in 3D is defined of points of the space $S^2 \times \Bbb{R}$. So, one could generalize this to $S^n \times \Bbb{R}^{n-1}$, or - as Joe did - $S^{n-1} \times \Bbb{R}$, or, what I would consider most general: $$ S^m \times \Bbb{R}^{n-m}, \quad0 < m < n $$ which would be my interpretation of the space in which a n-dimensional hypercylinder is defined.