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:
$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$ ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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
$\alpha+\beta=\gamma\equiv a^2+b^2=c^2$
$\...

Community wiki

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
Życzkowski, Karol; Sommers, Hans-Jürgen, Hilbert-Schmidt volume of the set of mixed quantum states, J. Phys. A, Math. Gen. 36, No. 39, 10115-10130 (2003). ...

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\...

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 ...

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 ...

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 ...

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<\...

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 ...

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