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7 votes
1 answer
179 views

More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$

Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let $$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
Iosif Pinelis's user avatar
4 votes
1 answer
96 views

On the Gram matrix of $6$ unit vectors in $\Bbb R^3$

Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$? Remark 1: A numerical experiment suggests that $...
Iosif Pinelis's user avatar
11 votes
2 answers
425 views

Maximization of a cubic form over the $14$-dimensional sphere

For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number. Is it true that, given the condition $$\sum_{1\le i<j\le6}x_{ij}^2=1,$$ the sum $$\sum_{1\le i<...
Iosif Pinelis's user avatar
1 vote
1 answer
323 views

An inequality in four variables

Let $f(x,y)=\frac{10xy-(x+y)+1}{8xy-2(x+y)+5}$ and $g(x,y)=\frac{1}{4}\left[1+\frac{1}{3}(4x-1)(4y-1)\right]$. I want to prove that for any $0.5\le a\le b\le 1$ and $0.7\le c\le d\le 1$, it holds that ...
lchen's user avatar
  • 367
3 votes
1 answer
195 views

Positivity of real functions in two variables

Assume that $f_0,f_1,f_2$ are polynomial functions of degree two in two variables. This means that the $f_i$ are linear combinations with real coefficients of $x^2,xy,x,y^2,y,1$. Consider the function ...
Puzzled's user avatar
  • 8,998
5 votes
5 answers
623 views

Elementary inhomogeneous inequality for three non-negative reals

I need the following estimate for something I am working on, but I don't immediately see how to establish it. For $x, y, z \in \mathbb{R}_{\ge 0}$, show that $$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
BPN's user avatar
  • 543
7 votes
1 answer
371 views

An elementary inequality for three complex numbers

The following problem arose in asymptotic analysis of difference equations. Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have $$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
Iosif Pinelis's user avatar
14 votes
1 answer
440 views

Inequalities on elementary symmetric polynomials

I have recently come across the following result. Let $0 < d \leq n$. Given any vector $x \in \mathbb{R}^n$ that satisfies $e_{d-1}(x) = 0$, show that $$|x_1 \cdots x_d| \leq |e_d(x)|$$ where $...
Nick R's user avatar
  • 1,187