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10 votes
0 answers
365 views

diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
Mikhail Katz's user avatar
  • 16.6k
6 votes
1 answer
388 views

Covering number estimates on closed Riemannian manifolds

Let $(M^n,g)$ be an $n$-dimensional compact and connected Riemannian manifold with sectional curvature bounded above and below by $c,C$. Is it possible/known how to express the external covering ...
Carlos_Petterson's user avatar
6 votes
1 answer
254 views

Triangulations of convex surfaces

Let $M$ be a smooth closed positively curved surface in Euclidean 3-space, $T$ be a geodesic triangulation of $M$, and $E$ be the edge graph of the convex hull of vertices of $T$. It is easy to see ...
Mohammad Ghomi's user avatar
2 votes
1 answer
159 views

Do all compact manifolds admit geodesic tiling

Let $M$ be a compact Riemannian manifold. I'll call a set of non-empty subsets $C_1,\dots,C_N$ a geodesic tiling of $M$ if: Each $C_n$ is closed (geodesically) convex hull of a finite number of $\{...
ABIM's user avatar
  • 5,405
1 vote
0 answers
111 views

Maximizing the minimum curvature of a convex shape with a given volume in higher dimensions

Given any $d$-dimensional convex shape $S$ in the Euclidean space with $d\gg 1$, let $K_{\min}(S)$ be the minimum value of the Gaussian curvature of its boundary. Question: What is the maximum value $...
Penelope Benenati's user avatar
0 votes
0 answers
125 views

Naming convention for different type of triangulations

When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general ...
Kregnach's user avatar
  • 183