Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,403 questions
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What is known about polyhedra nets that allow overlapping?
It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any ...
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Isometric embedding of a positively curved polyhedral surface
Suppose you have a 2-dimensional polyhedral surface with specified lengths for the edges so that the vertices all have positive curvature. I believe this has a unique isometric embedding into 3-...
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Topological embeddings of non-compact, complete metric spaces
Given a completely metrizable space, say that it has property X if it can be embedded in some metric space such that its image is not closed. For example, the real line R can be embedded, ...
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Maximal area coverable by $k$ disjoint isosceles triangles contained in a triangle of area 1.
Given a triangle $\Delta$ of unit area, how much area can always be covered by $k$ isosceles triangles contained in $\Delta$ and intersecting at most at their boundaries?
The answer is easy for $k=1$....
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Name for an inequality of isoperimetric type
I want to know if the following fact has a standard name and/or reference
Let $X$ be a subset of $\mathbb R^2$ and $B$ be a disc of the same area as $X$.
Set $X_\epsilon$ to be the $\epsilon$-...
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Diameter of a circle in an embedded Riemannian manifold
This question was inspired by an answer to the "Magic trick based on deep mathematics" question. I wanted to post it as a comment, but I ran out of characters! I'm sure there must be a collection of ...
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Hausdorff measure question
Say we have some compact metrisable topological space $X$ with a measure $\mu$ defined on the Borel sets of $X$. Then is there some way to determine whether $\mu$ is the Hausdorff measure associated ...
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Is a "contraction space" always complete?
Some of the fundamental results in analysis (inverse function theorem, existence and uniqueness of solutions to ODEs) have slick proofs using the idea of a contraction. So, it seems plausible to me ...
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Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets
Given two $n-$dimensional convex compact sets $A,B$, we define $d(A,B)$ as $\log({\mathrm{Vol}}(\alpha_2(A)))-\log(\mathrm{Vol}(\alpha_1(A)))$ where $\alpha_1,\alpha_2$ are two affine bijections such ...
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Optimizing finite-length approximations to space-filling loops
Take a loop in the unit disk D^2, with length l where length is defined as the supremum of the lengths of piecewise linear approximations. What is the smallest r such that every radius-r subdisk of D^...
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Can the circle be characterized by the following property?
In the Euclidean plane, is the circle the only simple closed curve that has an axis of symmetry in every
direction?
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In a locally CAT(k) space, does uniqueness of geodesics imply the lack of conjugate points?
A complete, simply connected Riemannian manifold has no conjugate points if and only if every geodesic is length-minimizing. I just realized that I don't know whether the same is true for a locally ...
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How far can the analogy between a Cayley graph and a symmetric space be pushed?
If $G$ is a finitely group and $S$ a finite symmetric set of generators, the associated Cayley graph, then $x \mapsto x^{-1}$ gives rise to a geodesic symmetry $i$ at the identity:
If $g=s_1^{e_1}\...
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Generalizing cosine rule to symmetric spaces
The sine and cosine rules for triangles in Euclidean, spherical and hyperbolic spaces can be understood as invariants for triples of lines. These invariants are given in terms of the distance (both ...
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Existence of Fermi coordinates on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold, $p$ a point on the manifold and $v \in T_p M$. Let $\gamma$ be the geodesic starting at $p$ in the direction $v$. There exists a time $t_f$ such that there ...
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Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?
This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a ...
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Are Bregman divergences quasi-convex?
Given a convex set S ⊂ ℝd and an appropriately differentiable convex function f: S → ℝ, a Bregman divergence Bf(x, y) = f (x) - f (y) -⟨x- y , ∇f (y)⟩ for x, y &...
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Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
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a general theory of configurations?
Once I found by accident an article by MacPherson: "Classical projective geometry and modular varieties", in "Algebraic analysis, geometry, and number theory" (Baltimore, MD, 1988), whose introduction ...
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When do two holonomy maps determine flat bundles that are isomorphic as just bundles (w/o regard to the flat connections)?
Suppose we have a surface S (although the question might make as much sense in higher dimensions) and a topological group G. The data of a flat vector bundle on S (up to isomorphism) is the same as a ...
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Is Level set of Regular functions in Alexandrov spaces again an Alex. space?
Let $X^n$ be an Alexandrov space, and $f: X^n\to \mathbb R^k$ a regular map, does the level set necessary be an Alexandrov space?
In my mind, the intrinsic metric on the level set is 'comparable' to ...
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Are isometries the only geodesic preserving maps in a CAT(0)-space?
Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with ...
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Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
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Maximizing Sparsity in l1 Minimization?
Consider the optimization problem
$$\min_x ||Ax||_1 + \lambda||x-b||^2,$$
where $A \in \mathbb{R}^{n \times n}$, $x,b \in \mathbb{R}^n$ and $\lambda$ is strictly greater than 0. (This problem is ...
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Why is 3 a bad constant in the Vitali covering lemma?
Hi,
recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...
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Algebraicity of the completion of a field? Finiteness?
At the end of my 8410 class today (see http://alpha.math.uga.edu/~pete/MATH8410.html if you care), one of my students asked me the following very interesting question:
Let $(K,|\ |)$ be a normed field,...
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Which changes of metric fix all open balls of a metric space?
In an earlier question, I was interested in counting the number of metric spaces on N points, where I considered two metric spaces to be the same if they had the same collection of open balls. Two ...
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What is the "right" universal property of the completion of a metric space?
I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes:
Let $X$ be a metric space. We all know that $X$ admits a completion, which is a complete metric ...
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Why is GL(n,C)/U(n) a CAT(0) space?
The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg ...
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Minimum-area bounding quadrilateral algorithm
There are a few algorithms around for finding the minimal bounding rectangle (OBB) containing a given (convex) polygon.
Does anybody know about an algorithm for finding a minimal-area bounding ...
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Variational characterization of curvature?
Consider a surface $S$ smoothly embedded in $\mathbb{R}^3$. Classically, the (Riemannian) curvature of $S$ is described by the second fundamental form, which is constructed from partial derivatives ...
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Zeta function for curves in a manifold
Motivation
In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = \text{log}\,p$. This is so you can express the zeta function as
$$ \...
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distance regular metric spaces
A metric space (V,d) will be called distance regular if for every distances a>0, b, c a nonnegative integer p(a,b,c) is defined, so that whenever d(B,C)=a, there are precisely p(a,b,c) points A ...
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Number of independent distances between n points in d-dimensional Euclidean space?
There are $\binom{n}{2}$ distances between $n$ points in $\mathbb{R}^d$. Not all of them can be chosen freely if $n$ exceeds the number $n_d = d + 1$. If $n = n_d$ we obviously have $\binom{d+1}{2}$ ...
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Simplicial and cubical decompositions of low valence
Every surface can be triangulated in such a way that at most 7 trianlges meet at one vertex. Every surface can be decomposed in squares such that at every vertex at most 5 suqares meet. For surfaces ...
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If a quadratic form is positive definite on a convex set, is it convex on that set?
Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...
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Combinatorial distance ≡ Euclidean distance
Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds:
disteuclidean(vi, vj) = f(distcombinatorial(vi, vj))
with ...
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Geometry of the multilagrangian Grassmannian
Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$.
Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
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what was Hilbert's geometric construction in his 17th problem?
Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
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What is the max number of points in R^3, interconnected by generic curves?
The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of it....
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Characterization of Riemannian metrics
This is probably an insanely hard question, but given an abstract metric space, is there some way to determine whether it's a manifold with a Riemannian, or more generally a Finslerian, metric? If ...
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Systems of conics
It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
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Tropical mathematics and enriched category theory
Is there a connection between tropical mathematics and the Lawvere enriched category theory approach to metric spaces? I guess I will give a partial answer to this below, but I mean can they be ...
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Minkowski sum of small connected sets
Suppose that the convex hull of the Minkowski sum of several compact connected sets in $\mathbb R^d$ contains the unit ball centered at the origin and the diameter of each set is less than $\delta$. ...
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Upper bound on the area of a midpoint pentagon?
Starting with a convex pentagon P, we define the "middle polygon" Q, whose vertices are the middle points of the sides of the initial pentagon. The ratio between the areas of this polygons seem to ...
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How to smootly interpolate between möbius transformations?
If you have two Möbius transformations represented as:
$f(z) = \frac{az + b}{cz + d}$
$g(z) = \frac{pz + q}{rz + s}$
where $a, b, c, d, p, q, r, s, z \in \mathbb{C}$
Is it possible to derive a ...
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