# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Bisections of the unit line inequality

Let $x_1 \in (0, 1)$. Iteratively define intervals $I_1,I_2,...$ and points $x_2,x_3,...$ by: $I_k$ is the longest sub-interval of $(0, 1)$ not containing any of the points $x_i , 1 \leq i \leq k$, (...
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### Prove that $\frac{1}{x^{1+\epsilon}}<\frac{1}{x(\log x)^p}$

Given $p>0$, $\epsilon>0$, prove that $\displaystyle \frac{1}{x^{1+\epsilon}}<\frac{1}{x(\log x)^p}$ for sufficiently large $x$. If $p\leq \epsilon$, then $(\log x)^p\leq x^p\leq x^\epsilon$...
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### When is the following trace inequality valid?

I have $A = A^T$ (and can have any real eigenvalue) and $B = B^T \succeq 0$ and want to know if the following holds $$trace(AB) \leq 0 \iff \lambda_{max} (AB) \leq 0$$ I know that the matrix $AB$ ...
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### Proving triangle inequality using complete-linkage between clusters and arbitrary dissimilarity measure

Assuming a dissimilarity measure d satisfies the usual properties, I need to prove that complete linkage ( i.e. d(A,B)=maxx∈A,y∈B{d(x,y)} ) either satisfies or does not satisfy the triangle inequality ...
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### What is the best time complexity of checking the inequality $a_1x_1 + \cdots + a_mx_m \le K$ to have a non-negative integer solution?

Consider $$a_1x_1 + \cdots + a_mx_m \le K$$ with $a_1, a_2, \ldots , a_m$ and $K$ being integers. I only need to know if the inequality has an integer solution or not. It means that there is ...
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### Can we say that $\prod_{i=1}^n (1-x_i)\ge 1-\sum_{i=1}^n x_i,\ \forall n\in \mathbb{N},\ \forall x_i\in [0,1)$?

The statement is easy to see to be true for $n=2,3$. However, what to do for general $n\in \mathbb{N}$? I am having this feeling that this should be a very trivial/well studied thing, but I am afraid ...
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### Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$

I got stuck on this inequality for a day. If $p$ is positive integer, then the problem becomes too easy, but I can't find how we deal with the general case when p can be any positive number. Can ...
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### Bernoulli inequality application

On some level of math in school we learn about Bernoulli's inequality. Proof of its correctness is very common in textbooks as exercise, when we learn mathematical induction. Is Bernoulli's ...
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### Israel tst 2011 geometrical inequality

Inside an equilateral triangle of area $S$ lies a point, whose distances to the vertices are $x,y, z$. Prove that $xy + yz + zx \geq \frac{4}{\sqrt{3}} S$ I haven't got any idea yet. But I guess ...
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### Combinatorical problem [on hold]

$k$ is a natural constant.Determine $x,y,z$ knowing that $\binom{z+k}{x+y} + \binom{z}{x} \le k$ and $2x+y \le z$.
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### Meaning of $Ax \leq b$

I continue to come across $Ax \leq b$ or $Ax= b$ in optimization problem, but I am having trouble interpreting the meaning of this. Does this have a similar meaning to the following (Cramer's Rule) ...
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