Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
157 questions from the last 365 days
6
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1
answer
218
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A too good to be true moment inequality for empirical processes
I have a question regarding an inequality that I obtained which seems to be too good to be true.
Consider a sequence $(X_i)_{i\leq N}$ of independent and identically distributed r.v.s. with law $\mu$ ...
- 273
32
votes
2
answers
715
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Is the expected distance between two random interior points of a convex body always at most that of two random boundary points?
Let $K \subset \mathbb{R}^2$ be a convex body. Define two quantities:
Interior mean distance. Let $X, Y$ be independent and uniformly distributed in $K$. Set
$$\Delta(K) \;=\; \mathbb{E}\,\|X - Y\|.$$...
- 1,012
-4
votes
0
answers
118
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Does the expression (9/2)π³ − √(2π) + 4/(9π³) appear in known mathematical literature? [closed]
I am working with a bilateral crossing geometry — two pyramids base-to-base at θ = π/8 — and the following expression arises naturally from the dimensional structure of the cascade:
α⁻¹ = (9/2)π³ − √(...
- 3
30
votes
3
answers
3k
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Project Hail Mary, question? (Spoiler)
I just watched the movie Project Hail Mary yesterday, and I have a question about the ending of the movie.
(Spoilers)
At the end, they show the alien nicknamed “Rocky” in a protective clear polyhedral ...
- 72k
3
votes
1
answer
109
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How can I find the defender movement angle that minimizes the quarterback’s safe region in a rectangle?
I am studying a geometric optimization problem in the plane.
Let $R=[0,W]\times[0,H]$ be the rectangle with corners $(0,0)$, $(W,0)$, $(W,H)$, and $(0,H)$.
Let the quarterback and defender start at $q=...
- 31
7
votes
1
answer
280
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Source, if there is one, for this short proof of baby Gauss-Bonnet (total angle defect of a polyhedron $=4\pi$)?
The proofs I've found proceed by triangulating and then appealing to $V-E+F=2$.
I'd like to know if this easy strategy is familiar:
Proof. Define the hoped-for angle at a vertex to be 360° if interior,...
- 17.9k
4
votes
0
answers
126
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Classifying regular polyhedra starting from a bad definition
I'm currently writing a book for advanced high school students. I cover classical topics such as the classification of regular polyhedra (in 3d). I started from the following (bad) definition: "a ...
3
votes
1
answer
607
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A claim on wrapping 3D solids with a sheet of paper
Ref 1: How big a box can you wrap with a given polygon?
Ref 2: 'Trapping' 3D regions with sheets of paper
We look at wrapping 3d solid bodies with any given sheet of paper which is a polygonal ...
- 7,485
1
vote
0
answers
55
views
Topology of metric balls in non-compact surfaces of non-negative Gauss curvature
Let $M$ be a complete non-compact Riemannian surface of non-negative Gauss curvature, and let $B(x,r)$ denote an open metric ball in $M$. What can be said about the topology of $B(x,r)$?
More ...
- 23.3k
4
votes
0
answers
174
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Replacing the diameter with Gaussian widths in Talagrand's $\gamma_2$-functional
Let $(T,d)$ be the simplex of $\mathbb{R}^m$ endowed with the Euclidean distance. We define the Gaussian width as the function $w(T)=\mathsf{E}\sup_{t\in T}\langle t,g\rangle$ where $g=(g_i)_{i\leq m}$...
- 273
2
votes
0
answers
101
views
A bound on Talagrand's functional $\gamma_2(S,d)$ for subsets $S$ of the simplex in $\mathbb{R}^m$
For an arbitrary metric space $(T,d)$, for any $S\subset T$, we write $\Delta(S)=\Delta(S,d)$ to denote its diameter, so $\Delta(S)=\sup_{s,t\in S}d(s,t)$. An admissible partition sequence $(A_n)_{n\...
- 273
4
votes
0
answers
232
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Geometric lemma in Mockenhaupt-Seeger-Sogge
I've been trying to read the paper of G. Mockenhaupt, A. Seeger & C. Sogge, Wave Front Sets, Local Smoothing and Bourgain's Circular Maximal Theorem, 136 Ann. Math. 207 (1992).
In estimating a ...
- 343
25
votes
4
answers
2k
views
Intuitive explanation why the radii converge in this sequence of eights?
In the diagram, circles of the same color are congruent.
I have a nonintuitive proof that the radii converge. Is there an intuitive explanation?
- 5,360
5
votes
2
answers
428
views
Generalized Buffon's needle-type problem
Suppose I have $n$ line segments in the unit square, and they are positioned in such a way that if I sample a circle (or, say, square) of given area $A$ uniformly at random (e.g. sample the center ...
- 4,263
4
votes
0
answers
92
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What is the functorial connection between metric polyhedron and homotopy class of topological space?
In Mikhail Gromov's famous book "Metric Structures for Riemannian and Non-Riemannian Spaces", I see the following sentences in the introduction:
Once can functorially associate an infinite ...
- 363