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2
votes
1answer
209 views

another diameter-perimeter-area inequality

Recently I learnt that $$ \inf\frac{diam(C)(per(C)-2diam(C))}{area(C)}=0$$ where the infimum is taken over all plane convex bodies $C$ (say, with non-zero area). In other words, there is no ...
1
vote
1answer
212 views

The largest circle that encloses no points on a plane with points placed at $N$ random coordinates

I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in ...
11
votes
0answers
366 views

Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
7
votes
1answer
217 views

Smoothing of piecewise Euclidean Riemannian metrics

Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...
2
votes
0answers
93 views

Isometric decomposition

Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others an isometry?
2
votes
1answer
189 views

A quantitative version of Straszewicz's theorem?

Let $C$ be a compact convex subset of Euclidean space. Recall that $x\in C$ is an exposed point of $C$ if there is a plane $P$ such that $P\cap C = \{x\}$. It is obvious that exposed points are ...
4
votes
0answers
217 views

Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the variational problem $$ ...
1
vote
2answers
217 views

Enlcosing a set of ellipses within one ellipse

Hello, Is there an algorithm that takes in a set of ellipses and gives back and ellipse that encloses the set?
4
votes
0answers
151 views

Convex bodies with symmetric shadows.

Theorem. If all orthogonal projections of a convex body $K \subset \mathbb{R}^n$ onto $2$-dimensional subspaces have a center of symmetry, then $K$ has a center of symmetry. This is a classic result ...
1
vote
0answers
89 views

Recover a pixilated image using a camera that collects sparse, regular samplings of pixels

Say I have some two-dimensional black and white image, which we can represent as a set of coordinates $(c_1, ..., c_N) \in C$ for the placement of dark/black pixels on a bounded rectangular plane with ...
4
votes
0answers
127 views

Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?

Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...
3
votes
1answer
325 views

What is the expected value for this

If there are $8$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the open interval $\left(0,1\right)$, what is the expected largest size of ...
9
votes
1answer
432 views

A generalization of intermediate value theorem on R^k.

Let $f:[0,1]\to\mathbb R^k$ be a continuous function with $f(1) = \overrightarrow 0$. Is it true that there always exist $k$ points $0 \le a_1 \le a_2 \le \ldots \le a_k \le 1$ such that $\sum_{i=1}^k ...
6
votes
0answers
189 views

Families of triangulations of polygons in the plane

Let $P$ be a polygon in the plane. An "efficient" triangulation of $P$ is one that introduces no new vertices. We require that all introduced edges be straight and inside $P$. Every polygon in the ...
1
vote
1answer
239 views

AC and Euclidean Geometry [closed]

It there any relation between the axiom of choice and Euclidean Geometry ?? I mean what are the known statements, theorems or results in euclidean geometry that are dependent on AC ?? (this question ...
1
vote
1answer
240 views

Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form. Some background may be ...
1
vote
1answer
264 views

Minimizing the Perimeter of a polyomino

Imagine I generate some polyomino (http://en.wikipedia.org/wiki/Polyomino) with $N$ unit squares, under the constraint that I want to maximize the number of shared edges between unit squares, or ...
1
vote
1answer
156 views

Orbits of Product Lie Groups Action

Hi to all, Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...
5
votes
3answers
258 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) ...
11
votes
1answer
276 views

Polyominoes with double contact

Here is a problem which arose from an earlier question. I'll change the terminology but not the question: A polyomino is a region with a connected interior made by joining one or more unit squares ...
1
vote
1answer
243 views

Frustrating the number of possible common edges between two connected components composed of square Penrose tiles

Imagine I have two bags of square and planar unit square tiles, with Penrose-like "nodules" on their edges s.t. two tiles can only be placed together if their edges are flush (i.e. if the two vertices ...
23
votes
2answers
682 views

The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square? By "nonoverlapping" I mean: not sharing an interior point. By "touch" I mean: sharing a boundary point. ...
4
votes
1answer
173 views

Extreme rays in the cone of (semi)metrics

How many extreme rays are there on the polytopal cone formed by all semimetrics on a set with $n$ elements? Some background. Given a set $X$ with $n$ elements, the set of all semimetrics $d:X \times ...
2
votes
1answer
185 views

What is the Sequence that Maximizes this Distance?

I have posted this question here without answer. Maybe I can get some light here. Suppose we are given $n$ segments $l_1,...,l_n$ in $\mathbb{R}^2$ such that $|l_i|=i,\ \forall\ i=1,...,n$, where ...
14
votes
3answers
497 views

Smallest square to wrap a cylinder

Suppose you need to gift-wrap a cylinder (e.g., a can of tennis balls, or a large candle) of height $h$ and radius $r$. Here wrap is the natural sense of covering the surface area of the cylinder ...
7
votes
0answers
187 views

From convex geometry to contact topology

Here is a problem in contact topology that was suggested by Petya's answer to this mathoverflow question of mine. Let $S^* \mathbb{R}^n$ be the space of cooriented contact elements of $\mathbb{R}^n$. ...
0
votes
1answer
201 views

Smooth maps transverse to a foliation

Let $M$ and $N$ be smooth manifolds and let $S$ be a submanifold of $N$ ($\dim S < \dim N$). Let $\mathfrak S$ be a foliation of $S$. We say that a map between $M$ and $N$ is transverse to ...
2
votes
2answers
224 views

What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of subsets in a subdivision of the set of $30$ points into subsets such that all the points in each subset are on the boundary of the ...
11
votes
2answers
392 views

A problem on convex geometry

Consider a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior. Although the body is not necessarily symmetric, let us say that two points in its boundary $\partial K$ are ...
26
votes
3answers
972 views

“Softness” vs “rigidity” in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ ...
8
votes
1answer
347 views

Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$. Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if there is a continuous rotation of $P$ such that its orthogonal-projection shadows ...
1
vote
1answer
236 views

Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of $V\bigotimes \mathbb{C}$ ?

We say $L< (V\oplus V^{*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$ Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal ...
0
votes
1answer
212 views

Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a convex polygon in the plane and $v_{m+1}$ be a vertex in the interior of the convex polygon. Connect ...
2
votes
2answers
270 views

Convex upper bound on a linear-fractional function

I have a function of the form $f(x,y) = \frac{x}{c+y}$ where $c$ is a positive constant, $c \ge x \ge 0$, and $y \ge 0$. I would like to find a convex upper-bound for this function. Is there a ...
2
votes
1answer
277 views

Manhattan distance vs. absorption time on an unbounded integer lattice

Imagine I have unbounded $d$-dimensional integer lattice where I take two vertices, $v_a$ and $v_b$, separated by a fixed Manhattan distance $L$, and I release a random walker at $v_a$ and allow for ...
9
votes
3answers
331 views

Mutually tangent ellipsoids in 3 space

I recently heard a claim that for any n, it is possible to arrange n ellipsoids in 3 space such that each pair of ellipsoids is kissing. Is this true, and if so, how? Edit: By kissing, I mean that I ...
5
votes
2answers
118 views

Abstract characterization of polygonizations

Consider a polygonization of the plane by convex polygons of a given minimal size that meet edge-to-edge and vertex-to-vertex. What's the “official” name of such a polygonization? Such ...
4
votes
3answers
257 views

lines through A_n reflection arrangement and permutations

(updated; apologies for way too much room left for interpretation in the original post) Let $\mathcal{A} =A_{n-1}$ be the $A_{n-1}$ arrangement in $\mathbb{R}^{n}$, i.e. the set of hyperplanes ...
5
votes
4answers
2k views

Proof without words for surface area of a sphere [closed]

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the ...
11
votes
2answers
869 views

There are two points on the Earth's surface that … ?

At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...? What is the strongest, most impressive statement one can make here? The ...
2
votes
2answers
535 views

Has the notion of “space” been reconsidered in 20th century?

The original title, "has the bases of geometry been reconsidered in 20th century" of this question refers to Riemann's paper "On the Hypotheses which lie at the Bases of Geometry", an English version ...
1
vote
1answer
282 views

Leray Spectral Sequence

Let $f:X\to Y$ be a smooth map between paracompact differential manifolds $X$ and $Y$. Let $U$ be an open and dense subset of $Y$. For any $y\in U$, let $f^{-1}(y)=F$ be a generic fiber that is a ...
12
votes
1answer
280 views

The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection of rectifiable curves in the plane with the property that every unit-length segment meets at least one curve in at least one point. Call such a ...
2
votes
1answer
735 views

Subtract Rectangle from Polygon

I'm looking for an algorithm that will subtract a rectangle from a simple, concave polygon and return a remainder of polygons. If the rectangle encloses the polygon, the remainder is null. In most ...
8
votes
11answers
2k views

Textbook for undergraduate course in geometry

I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry ...
8
votes
2answers
762 views

About MF Atiyah and R Bott's 1983 paper

I am a theoretical physics major student working on string theory. I want to understand the work of MF Atiyah and R Bott, "The Yang-Mills equations over riemann surfaces" . What kinds of mathematical ...
1
vote
1answer
275 views

Properties of the minimum circumscribing circle for points sequentially placed within a circle of radius $r$ of previously placed points on a 2D plane

Imagine I perform the following procedure: [1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$. [2] At time point $t_2$, I center a ...
3
votes
1answer
401 views

Maximizing the number of lattice points in a circle of radius $r$ placed on a lattice

I have a circle of radius $r$, and I wish to place this circle of a $Z^2$ integer lattice or an $A_2$ hexagonal lattice s.t. I maximize the number of lattice points within or along the contour of the ...
2
votes
2answers
493 views

An exact counting solution for the number of points within a circle of radius $r$ centered on a lattice point in a $A_2$ hexagonal lattice

In a previous question: (The Gauss circle problem on a hexagonal lattice) I asked for an analytic approximation for the number of lattice points in or along the contour of a circle centered on a ...
2
votes
2answers
281 views

Self-Intersection of closed curves

Supoose I have a closed curve $\gamma$ in the plane such that for any isometry $g$ of $\mathbb{E}^2,$ such that $g(\gamma)\neq \gamma,$ $\gamma$ intersects $g(\gamma)$ in at most two points. It ...