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Shinpei
Shinpei
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I would appreciate comments and references for the following question that I have been curious about.

Let X be a convex polyhedron in hyperbolic 3-space.

Let M be the medial axis of X.

Question: Is M collapsible?

It is easy to see that M is contractable. In the case of Euclidian 3-space, instead of hyperbolic 3-space, I think I have an elementary proof for the analogues statement. In hyperbolic space, I guess that the statement still holds, but I do not have a proof.

I would appreciate comments and references for the above question.

Definitions for the above question

The MEDIAL AIXS of X : consider a maximal round ball inscribed in X (maximal with respect to set inclusion), which at least has two points of tangency (on $\partial X$). Take the union of the centers of all such maximal balls inscribed in X. Then, this union is called the medial axis of X, and it is a polygonal complex, to be precise, after adding the 1-skeleton of the boundary of X.

M is COLLAPSIBLE: There is a strong deformation retract of M to a point that is a composition of certain simplicial homotopies, each of which reduces a single cell.

I would appreciate comments and references for the following question that I have been curious about.

Let X be a convex polyhedron in hyperbolic 3-space.

Let M be the medial axis of X.

Question: Is M collapsible?

It is easy to see that M is contractable. In the case of Euclidian 3-space, instead of hyperbolic 3-space, I think I have an elementary proof for the analogues statement. In hyperbolic space, I guess that the statement still holds, but I do not have a proof.

Definitions for the above question

The MEDIAL AIXS of X : consider a maximal round ball inscribed in X (maximal with respect to set inclusion), which at least has two points of tangency (on $\partial X$). Take the union of the centers of all such maximal balls inscribed in X. Then, this union is called the medial axis of X, and it is a polygonal complex, to be precise, after adding the 1-skeleton of the boundary of X.

M is COLLAPSIBLE: There is a strong deformation retract of M to a point that is a composition of certain simplicial homotopies, each of which reduces a single cell.

Let X be a convex polyhedron in hyperbolic 3-space.

Let M be the medial axis of X.

Question: Is M collapsible?

It is easy to see that M is contractable. In the case of Euclidian 3-space, instead of hyperbolic 3-space, I think I have an elementary proof for the analogues statement. In hyperbolic space, I guess that the statement still holds, but I do not have a proof.

I would appreciate comments and references for the above question.

Definitions for the above question

The MEDIAL AIXS of X : consider a maximal round ball inscribed in X (maximal with respect to set inclusion), which at least has two points of tangency (on $\partial X$). Take the union of the centers of all such maximal balls inscribed in X. Then, this union is called the medial axis of X, and it is a polygonal complex, to be precise, after adding the 1-skeleton of the boundary of X.

M is COLLAPSIBLE: There is a strong deformation retract of M to a point that is a composition of certain simplicial homotopies, each of which reduces a single cell.

Source Link
Shinpei
Shinpei
  • 133
  • 1
  • 1
  • 5

Collapsing the medial axis of a polytope

I would appreciate comments and references for the following question that I have been curious about.

Let X be a convex polyhedron in hyperbolic 3-space.

Let M be the medial axis of X.

Question: Is M collapsible?

It is easy to see that M is contractable. In the case of Euclidian 3-space, instead of hyperbolic 3-space, I think I have an elementary proof for the analogues statement. In hyperbolic space, I guess that the statement still holds, but I do not have a proof.

Definitions for the above question

The MEDIAL AIXS of X : consider a maximal round ball inscribed in X (maximal with respect to set inclusion), which at least has two points of tangency (on $\partial X$). Take the union of the centers of all such maximal balls inscribed in X. Then, this union is called the medial axis of X, and it is a polygonal complex, to be precise, after adding the 1-skeleton of the boundary of X.

M is COLLAPSIBLE: There is a strong deformation retract of M to a point that is a composition of certain simplicial homotopies, each of which reduces a single cell.