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

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 ...
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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 ...
• 9,758
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 ...
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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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