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3 votes

An inequality in an Euclidean space

It doesn't hold, although it looks quite rare. For example $$u:=\begin{pmatrix}1\\1\\1\end{pmatrix}, \quad v:=\begin{pmatrix}\;1\\\;1\\27\end{pmatrix}, \quad x:=\begin{pmatrix}-1\\\;\;6\\\;25\end{...
Claude Chaunier's user avatar
7 votes
Accepted

An inequality in an Euclidean space

Imagine that $v$ is (almost) equal to $(1,0,\ldots,0)$, then for $x=(x_1,\ldots,x_n)$, $\sum x_i=:na$, we should check that $(x_1-a,x_2-a,\ldots,x_n-a)\cdot(0,x_2,\ldots,x_n)\geqslant 0$, in other ...
Fedor Petrov's user avatar
1 vote

An inequality in an Euclidean space

There is nothing special about $\mathbf{u}=(1,\dots,1)$. I work with an arbitrary vector in $(0,\infty)^n$. Claim) Let $n\geq 3$ and $\mathbf{u}=(u_1,\dots,u_n),\mathbf{v}=(v_1,\dots,v_n)\in (0,\...
KhashF's user avatar
  • 2,867
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
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
  • 2,867
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
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
2 votes

Why do all incidence theorems follow from Pappus' theorem?

There's a recent talk by Sergey Fomin about this topic exactly - https://www.youtube.com/watch?v=kU5PazTfImo Abstract: We show that various classical theorems of real/complex linear incidence geometry,...
Nick Ulyanov's user avatar
2 votes

Inscribing one regular polygon in another

Here is a proof for large enough coprime $m$ and $n$. I use some basic properties of cyclotomic polynomials, be free to ask for details if needed. In particular, I use that the sum of roots of $\Phi_k$...
Fedor Petrov's user avatar

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