Loading…
Rigidity and Volume Preserving Deformation on Degenerate Simplices
Given a degenerate ( n + 1 ) -simplex in a d -dimensional space M d (Euclidean, spherical or hyperbolic space, and d ≥ n ), for each k , 1 ≤ k ≤ n , Radon’s theorem induces a partition of the set of k -faces into two subsets. We prove that if the vertices of the simplex vary smoothly in M d for d =...
Saved in:
| Published in: | Discrete & computational geometry 2018-12, Vol.60 (4), p.909-937 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Given a degenerate
(
n
+
1
)
-simplex in a
d
-dimensional space
M
d
(Euclidean, spherical or hyperbolic space, and
d
≥
n
), for each
k
,
1
≤
k
≤
n
, Radon’s theorem induces a partition of the set of
k
-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in
M
d
for
d
=
n
, and the volumes of
k
-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all
k
-faces; and this property still holds in
M
d
for
d
≥
n
+
1
if an invariant
c
k
-
1
(
α
k
-
1
)
of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant
c
k
(
ω
)
we discovered for any
k
-stress
ω
on a cell complex in
M
d
. We introduce a characteristic polynomial of the degenerate simplex by defining
f
(
x
)
=
∑
i
=
0
n
+
1
(
-
1
)
i
c
i
(
α
i
)
x
n
+
1
-
i
, and prove that the roots of
f
(
x
) are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case. |
|---|---|
| ISSN: | 0179-5376 1432-0444 |
| DOI: | 10.1007/s00454-017-9956-x |