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Results tagged with alexandrov-geometry
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user 1441
Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).
28
votes
8
answers
5k
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Convex hull in CAT(0)
Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset.
Is it true that convex hull of $K$ is compact?
Comments:
Convex hull of $K$ = intersection of all closed convex set …
- 45k
17
votes
1
answer
931
views
Minimizing geodesic on a convex surface
Let $\Sigma$ be a smooth convex surface in Euclidean 3-space
and $\gamma$ be a unit speed minimizing geodesic in $\Sigma$.
Assume that for some $a < b < c$, we have
$$\gamma'(a)=\gamma'(b)=\gamma'(c) …
- 45k
16
votes
5
answers
1k
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(1-Lipschitz) + (length-preserving) = isometry
I am looking for an elementary way to prove the following theorem.
Theorem. Let $\alpha$ and $\beta$ be two simple convex closed curves in $\mathbb R^2$.
Assume
$$\mathop{\rm length} \alpha=\ …
- 45k
12
votes
1
answer
380
views
Connecting Lemma in the Alexandrov's existence theorem.
At the moment I am polishing my lecture notes which in particular cover Alexandrov's existence theorem.
Denote by $\mathbf{P}_k$ the space of isometry classes of polyhedral
metrics on the $\mathbb S …
- 45k
9
votes
0
answers
176
views
Involution of 3-sphere
Suppose that a closed geodesic $\gamma$ is the fixed-point set of an isometric involution on $(\mathbb{S}^3,g)$. Assume that sectional curvature of $g$ is at least $1$.
Is it true that $$\mathrm{leng …
- 45k
8
votes
0
answers
272
views
Generalized flag complex?
Assume we glue an $n$-dimensional simplicial complex $K$
from copies of an $n$-simplex $\Delta$ with fixed spherical metric.
We may think that $\Delta$ has colored vertices
and we glue so that the col …
- 45k
6
votes
0
answers
134
views
Nearby convex set in a nearby space
Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$.
Is there a convex set $K'\subset X'$ that is close to $K\subset X$?
Two spaces $X$ and $X'$ a …
- 45k
6
votes
1
answer
349
views
Harmonic maps are light
Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map
and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected?
I hope that the answer is yes.
But actually I …
- 45k
2
votes
0
answers
80
views
Nested convex hulls in Hadamard manifold
Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood.
Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$?
Commen …
- 45k
1
vote
1
answer
108
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Alexandrov's uniqueness theorem in Minkowski spacetime
Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$.
Each face of $P$ comes with induced metric tensor,
if the face is space-like, then it is euclidean metric;
every time-like face is isometric t …
- 45k