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19 votes

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

The area, call it $K$, is indeed maximized when the quadrilateral is cyclic, in which case $K$ is given by Brahmagupta's remarkable generalization of Heron's formula: $K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ ...
Noam D. Elkies's user avatar
17 votes

Packing an upwards equilateral triangle efficiently by downwards equilateral triangles

OK, posting then. I prefer to think of triangles pointing to the right in the triangle pointing to the left. Let $\delta=e^{-\sqrt{\log 1/\varepsilon}}$. For each small triangle $T$, let $I$ be the ...
fedja's user avatar
  • 60k
15 votes

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

Another proof that the area is maximized when the quadrilateral is cyclic is the following: First we consider a cyclic quadrilateral with the given sides lengths (the existence of such a quadrilateral ...
Corentin B's user avatar
9 votes
Accepted

Packing an upwards equilateral triangle efficiently by downwards equilateral triangles

Ok I think this is an argument that the perimeter at least $\varepsilon^{-c}$ for some sufficiently small $c$. To avoid special cases where $\varepsilon$ is large, we use the convention that we count ...
Anders Martinsson's user avatar
5 votes

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

I argue that the expression for infimum is more complicated and cannot be given by a single formula unless certain inequalities involving $a,b,c,d$ are assumed. Let us work in the setting that OP ...
KhashF's user avatar
  • 2,857
4 votes

Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces

Even for coarse maps between Gromov-hyperbolic spaces $f: X\to Y$ there are neither reasonable upper nor lower bounds of the type $$ \psi_-((x,y)_z)\le (f(x), f(y))_{f(z)}\le \psi_+((x,y)_z) $$ (where ...
Moishe Kohan's user avatar
  • 9,758
3 votes
Accepted

Concentration of measure on spheres with respect to a unitary of trace approximately zero

This is indeed true. Assume WLOG that the matrices are diagonal. A useful way to handle the uniform measure on the sphere is that if you let $X_1,\ldots,X_n$ be iid (complex) Gaussians, then the ...
Marcus M's user avatar
  • 944
3 votes

Which theorems have Pythagoras' Theorem as a special case?

Dijkstra's generalization of the Pythagorean theorem Let a triangle have sides $a,b,c>0$ with corresponding opposite angles $\alpha,\beta,\gamma$. Then $\alpha+\beta=\gamma\equiv a^2+b^2=c^2$ $\...
3 votes

What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?

This question has been investigated in ┼╗yczkowski, Karol; Sommers, Hans-J├╝rgen, Hilbert-Schmidt volume of the set of mixed quantum states, J. Phys. A, Math. Gen. 36, No. 39, 10115-10130 (2003). ...
Guillaume Aubrun's user avatar
3 votes

Product of low dimensional Hausdorff measures

I assume that your measure is defined as $$ \mathcal H^s\otimes\mathcal H^r(S) := \inf\left\{\sum_{k=0}^\infty\mathcal H^s(A_k)\mathcal H^r(B_k)\middle|A\in\mathcal B(\mathbb R^n)^{\mathbb N},B\in\...
Pierre PC's user avatar
  • 3,074
3 votes
Accepted

Continuity of the volume function

I think you may have more restrictive hypotheses in mind, because in general this will be false due to behavior that may be obviously absent in the cases you want to consider. Anyway, here are two ...
Pierre PC's user avatar
  • 3,074
2 votes

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

The basic fact is that one can express the required extreme values of the areas in the generic case (our use of this term is explained below). The simple formulae involved can be found below. To do ...
crow's user avatar
  • 41
1 vote

Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon

This is about existence for both the max-min angle and the min-max angle problem. Existence of the optimiser is not immediately obvious for both, since the objective functions are combinations of ...
Pietro Majer's user avatar
  • 56.6k
1 vote
Accepted

Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon

This is about the issue of uniqueness, with a slight reformulation of the problem in view of an algorithmic approach. Given a (closed) half-plane, a segment $e$ of its boundary, and $0<\theta<\...
Pietro Majer's user avatar
  • 56.6k
1 vote
Accepted

A metric characterization of Hilbert spaces

Probably I.G. Nikolaev. See Theorem 10.10.13 in Burago, Dmitri; Burago, Yuri; Ivanov, Sergei "A course in metric geometry" [https://www.ams.org/books/gsm/033/][1] Hat tip to A. Eskenazis who ...
Manor Mendel's user avatar

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