Tagged Questions

99 views

Circumscribing simplex to convex body?

Q. Does every (perhaps smooth) compact convex body $K$ in $\mathbb{R}^d$ admit a circumscribing simplex, each facet of which touches (shares a point with) $K$? How about a circumscribing ...
478 views

A metric space of geometric shapes

My research involves geometric shapes in $R^2$, and I need a metric with several properties such as: Families of similar shapes, such as squares, are closed in this metric. Also more general ...
96 views

Star shaped sets with a midpoint

Suppose $U$ is an open subset of $\mathbb{R}^n$ which is star shaped with respect to $p\in U$. I'll call $p$ a midpoint of $U$ if for any line $\ell$ through $p$, the point $p$ is the midpoint of the ...
141 views

Areas of Triangles in (Non-Riemannian) Metric spaces?

I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this ...
181 views

Generalization of notion of convexity

I am searching for the correct term for the following, if it exists. A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x_1, x_2\in X$ such that there exists an arc of radius ...
211 views

Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...
82 views

Constructing measures with support in a given set

I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...
459 views

Is this knot invariant already treated somewhere in the literature?

Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$ We can turn $Y_K$ into a metric space by considering the distance induced by ...
137 views

Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes. For a given $\delta$, let $n_\delta$ be the number of faces ...
308 views

Reference request for an early theorem of Gromov

In his talk Misha Gromov- How does he do it, Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: Every manifold admitting a sequence of metrics such that the diameter ...
422 views

An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
140 views

Bounding the perimeter of a geodesic triangle in spaces of non-positive curvature

This is probably an easy question, but I don't know any Riemannian geometry and a literature search hasn't helped. Any help (e.g. providing a reference) would be greatly appreciated. For a triangle ...
101 views

Maximizing ratio volume/diameter^n by an affinity

Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$. It is clear that there is a constant ...
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Polyhedra with minimal edge length

Given a fixed volume and fixed surface area I would like to construct polyhedra that minimize the total length of the edges. This seems like a straight-forward problem to solve by brute force for ...
93 views

Definition of self-dual polytope

Given a d-polytope $P$ we define the c-dual polytope as $P^\ast = \{y\in R^d \mid x\cdot y\geq -c, \forall x\in P\}$. Then I say that a polytope is c-polar self-dual if $P=P^\ast$. I cannot find this ...
117 views

Circumscribed ellipsoid of minimum Hilbert-Schmidt norm

Let $K\subseteq \mathbb{R}^n$ be a full-dumensional convex body. The Löwner ellipsoid of $K$ is the unique ellipsoid of smallest volume containing $K$. My question is about a related object: the ...
114 views

Space of simple polygons on $n$-vertices as a set of points in $\mathbb{R}^{2n}$

A simple polygons in $\mathbb{R}^2$ with $n$ vertices can be mapped to elements in $\mathbb{R}^{2n}$ by list the coordinates of it's vertices. I expect there might be something interesting to study ...
265 views

Sphere - Symmetry and Triangulation [closed]

The sphere is symmetric with respect to any rotation. However, it loses this property as soon as it is triangulated. Are there sequences of triangulations that possess particular large symmetry groups ...
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Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown. I have already asked this question on ...
345 views

Oloid and sphericon: rolling develops entire surface

Wikipedia says that, "The oloid is one of the only known objects, along with some members of the sphericon family, that while rolling, develops its entire surface." Below are illustrations of ...
2k views

Who first realized that the shortest distance between two points is a straight line? [closed]

It's a bit like asking who invented the wheel, but perhaps there's something out there. This seems beyond the obvious crew: Euclid, Pythagoras, etc. Is there any evidence when this became common ...
434 views

Sets of evenly distributed points in the Euclidean plane

Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite? If the answer is yes, can ...
167 views

Three questions concerning lattice points on sphere surfaces

Pardon my ignorance of this topic. Q1. In which dimensions $d$ is it the case that, for every natural number $n$, there exists a sphere having exactly $n$ lattice points on it ...
190 views

Bi-spherical polyhedra

Bicentric polygons have been studied: a polygon all of whose vertices lie on its circumcirle, and whose incircle is tangent to every edge:   I have not been able to find a comparable literature ...
210 views

Passing C through a slot

Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through? i.e. which are the necessary and sufficient conditions for a planar compact set C to ...
156 views

Hales's fan associated with a polyhedron

In Hales's book (cited below), he associates what he calls a fan with any convex polyhedron in $\mathbb{R}^3$. I will not define his notion of fan, but let his figure (p.137) serve as a definition: ...
254 views

Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?

In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' ...
196 views

Divergence of geodesics in mapping class groups

I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the ...
463 views

Integer-distance sets

Let $S$ be a set of points in $\mathbb{R}^d$; I am especially interested in $d=2$. Say that $S$ is an integer-distance set if every pair of points in $S$ is separated by an integer Euclidean distance. ...
548 views

The Icosahedron Equation

$$1728 V^5 + F^3 = E^2 \;.$$ Can anyone point me to a concise, modern derivation and explanation of the significance of the icosahedron equation, more modern and concise than Klein's description in ...
152 views

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Optimal inspection path on a sphere

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand. ...
120 views

Optimal paintbrush geodesics

Let $S$ be a smooth, closed surface in $\mathbb{R}^3$, and $\gamma$ a geodesic segment on $S$, i.e., a finite-length piece of a geodesic. Define $\gamma(w)$ as all the points of $S$ within a distance ...
428 views

For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.) Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...
299 views

Reference for ultrametric spaces

I have a research project involving ultrametric spaces, and there are some facts that I use but have a hard time finding explicitely in the literature, although I know that some of them are folklore ...
750 views

reference for “X compact <=> C_b(X) separable” (X metric space)

I know (and am able to prove via Stone-Čech compactification) that the following is correct: Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued ...
265 views

Which metric spaces have this superposition property?

Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$. Say that a metric space $X$ has the superposition property (my terminology) ...
560 views

The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?            The above is a photo of foil I flattened to reuse. It might be ...
458 views

Egg-ovoid rolling down an inclined plane

I am seeking a mathematical analysis of an egg-ovoid rolling down an inclined plane, for pedagogical reasons. It is well-known folk lore that the shape of an egg prevents it from rolling away from ...
149 views

Compute generalized pentagram map

Hi, (This is my first question on MathOverflow! :-) Imagine you have a set of points $S = \{p_1, \ldots, p_n\}$ in $\mathbb{R}^d$, of which $t$ are "bad". I want to compute a "safe convex hull", ...
333 views

Isoperimetric inequality in complex hyperbolic space

Let $\mathbb{H}_\mathbb{C}^n$ be n-dimensional complex hyperbolic space. This space is a complex analog of hyperbolic space. It is isometric to the quotient of hyperboloid ...
2k views

Weitzenböck Identities

I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of ...
323 views

Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere $S$ in $\mathbb{R}^3$, sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$, with $\alpha < 1$ the fraction of ...
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Does this Banach manifold admit a Riemannian metric?

First, the question; after, the motivation. Consider 27.6 (pdf pp. 262-263) in The convenient setting of global analysis (AMS, 1997), and, in particular, the example given at the end of it, which ...
330 views

Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?

Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary. One way to define $\partial X$ is as the equivalence class of geodesic rays $\gamma(t), \gamma'(t)$ that remain within a ...