Tagged Questions
3
votes
1answer
44 views
Find min of $IA + IB + IC +ID$ in tetrahedron $ABCD$
Let the point $I$ in tetrahedron $ABCD$. Find $\min\{IA + IB + IC + ID\}$.
I can't solve this problem, even in the case ABCD regular. Please help
5
votes
4answers
120 views
Arithmetic mean is less than geometric mean (Spivak Calculus 3rd Chapter 2 Problem 22)
If $a_1, \ldots, a_n \ge 0$, the arithmetic mean $$A_n={a_1 + \cdots + a_n \over n}$$ and the geometric mean $$G_n = \sqrt[n]{a_1 \cdots a_n}$$ satisfy $G_n \le A_n$.
As a first step to prove this ...
2
votes
2answers
175 views
Finding the maximum of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$
If $a,b,c,d$ are distinct real numbers such that $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $ac=bd$.
Then how would we calculate the maximum value of ...
4
votes
4answers
91 views
Find minimum in a constrained two-variable inequation
I would appreciate if somebody could help me with the following problem:
Q: find minimum
$$9a^2+9b^2+c^2$$
where $a^2+b^2\leq 9, c=\sqrt{9-a^2}\sqrt{9-b^2}-2ab$
0
votes
2answers
50 views
Finding the minimum of $6a^3+9b^3+32c^3+\frac{1}{4abc}$ for positive $a,b,c$
If $a,b,c$ are real positive numbers. How to find the minimum for:
$$6a^3+9b^3+32c^3+\frac{1}{4abc}$$
0
votes
1answer
56 views
Prove the A-G-M Inequality using Lagrange multipliers.
I’m trying to prove the Arithmetic-Geometric-Mean Inequality (A-G-M) using Lagrange multipliers. For positive real numbers $ x_{1},x_{2},\ldots,x_{n} $, we want to show that
$$
(x_{1} x_{2} \cdots ...
1
vote
1answer
50 views
The general intution as to why the vector with the norm of 0 is chosen during the Cauchy-Schwarz inequality proof.
My professor mentioned that reason $t$ was chosen during the Cauchy-Schwarz inequality proof can also be seen when we minimize $\lVert x\rVert^2 +2t\langle x,y \rangle +t^2\lVert y\rVert^2$ over $t ...
12
votes
2answers
527 views
the least value for :$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$
For every $a,b,c$ non-negative real number such that:$a+b+c=1$ how to find the least value for :
$$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$$
9
votes
3answers
245 views
Find the maximum and minimum of $\sum_{i=1}^{n-1}x_ix_{i+1}$ subject to $\sum_{i=1}^nx_i^2=1$.
Find the maximum and minimum of
$$
\sum_{i=1}^{n-1}x_ix_{i+1}
$$
subject to
$$
\sum_{i=1}^nx_i^2=1
$$
for all $n\in\mathbb{N}-\{1,0\}$.
2
votes
1answer
47 views
Maximalization of a cubic puzzle
What is the maximal volume of a post package of length $L$, width $W$ and height $H$,
subject to the following restrictions:
$L+W+H \leq 90 $
$L \leq 60$, $W \leq 60$, $H \leq 60$
Intuitively I ...
1
vote
1answer
132 views
Simple explanation of Comb inequalities in TSP
A comb can be defined by a handle $H$ and a number of teeths $T_1,T_2,\dots,T_t$ such that:
$H,T_1,T_2,\dots,T_t \subseteq V$
$T_j \setminus H \neq \emptyset$ $\,\,\, \forall 1 \leq j \leq t$
$T_j ...
0
votes
2answers
32 views
Maximising an expression
We are to maximise $x^{2}y-y^{2}x$, where $x,y \in [0,1]$. I've tried using AM-GM to find another (easier to maximise) expression, which gave me $xy(x-y) \le \frac{1}{2}(x^{2}+y^{2}) (x-y)$ but that ...
0
votes
1answer
399 views
proving kantorovich inequality
Nevermind! I worked it out. This question seems really silly now.
I am working through the proof of the kantorovich inequality on pages 6 and 7 of the following lecture notes: u.mich optimisation ...
3
votes
1answer
122 views
Minimizing a function over two variables
Given two natural numbers $i$ and $p$ such that $0 < i \leqslant 2^p$, let
$$
\psi(p,i) := p - \alpha + 1 - \frac{1}{2^p}\left((2^p+i)\lg(2^p+i) - i\lg i - i + \alpha - \frac{2^p}{i+1} -
...
2
votes
1answer
88 views
Equality in the Isoperimetric Inequality
Stein and Shakarchi, in their book Real Analysis, the third volume of the Princeton Lectures in Analysis series, give a proof of the isoperimetric inequality for closed rectifiable curves in ...
7
votes
1answer
210 views
Maximum subset sum of $d$-dimensional vectors
This is a $d$-dimensional generalisation of the post Inequality with Complex Numbers. (See my comment under Robert Israel's answer.)
Generalising Potato's proof for $d$-dimensions, we can show the ...
5
votes
1answer
363 views
Least-squares left-inverse having smallest Frobenius norm
While trying to prove that the left-inverse of $A$ provided by the least-squares solution to $y=Ax$ has the smallest Frobenius norm, I am stuck at a point which I describe below:
Let $B$ be any ...
29
votes
4answers
771 views
AM-GM-HM Triplets
I want to understand what values can be simultaneously attained as the arithmetic (AM), geometric (GM), and harmonic (HM) means of finite sequences of positive real numbers. Precisely, for what points ...
2
votes
1answer
105 views
Maximise $L^q$ norm of a vector, for fixed $L^1$ and fixed $L^p$ norms
Consider a vector $x \in \mathbb R_+^n$ and $p,q \in \mathbb R$ such that $1<p<q$.
We fix $\sum \limits_{i=1}^{n}|x_i| = 1$ and $ \left(\sum \limits_{i=1}^{n}|x_i|^p \right)^\frac{1}{p} = ...
0
votes
2answers
304 views
Maxima of bivariate function
[1] Is there an easy way to formally prove that,
$$
2xy^{2} +2x^{2} y-2x^{2} y^{2} -4xy+x+y\ge -x^{4} -y^{4} +2x^{3} +2y^{3} -2x^{2} -2y^{2} +x+y$$
$${0<x,y<1}$$
without resorting to checking ...
1
vote
2answers
160 views
How to find the minimum value of $px+qy$ when $xy=r^2$?
The question says: "Find the minimum value of $px+qy$ when $xy=r^2$."
No information is given on $p,q,x,\text{and }y.$ However assuming the obvious I tried using this, but I am not able reduce it to ...
3
votes
2answers
60 views
a plausible maximum or minimum
Is the following statement true? Let $a_1\ge a_2\ge \cdots \ge a_n>0$, $b_1\ge b_2\ge \cdots \ge b_n>0$, then
$$\max\limits_{\sigma\in S_n}\;\;\prod\limits_{i=1}^n(a_i+b_{\sigma ...
3
votes
3answers
77 views
Prove that $(2-x)^nx^{n-1}$ decreases with $n$ for $0 <x<1$?
How can I show that:
$$(2-x)^nx^{n-1}$$ is decreasing with $n$ when $0<x<1$? I think this is generally true, but specifically I am concerned with $n$ as an integer $\geq 2$ and showing that the ...
4
votes
8answers
1k views
Maximizing the sum of two numbers, the sum of whose squares is constant
How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal?
This result is also true for ...
3
votes
6answers
274 views
Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$
If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$.
How to solve this one?
(Here's the source of inspiration for the problem.)
1
vote
1answer
122 views
Find the minimum value of $(\frac{x^n}{n} + \frac{1}{x})$ for $n \ge 4$
Find the minimum value of $(\frac{x^n}{n} + \frac{1}{x})$ for $n \ge 4$.
One possible approach could be by first writing $$ \left(\frac{x^n}{n} + \frac{1} {x}\right) = \left( \frac{x^n}{n} + ...
5
votes
1answer
106 views
Minimize and maximize length of a polygonal chain with certain boundary conditions
let $P_0,\ldots, P_k\in \mathbb{R}^2$ be a set of points. Furthermore let $\epsilon\in \mathbb{R}$.
Now I am trying to find non-trivial lower and upper bounds for
$$
\sum_{i=1}^k ...
1
vote
1answer
67 views
The lower bound of the product between two variables
I wonder how I can determine the minimum of the product between variables $x$ and $y$ (in terms of $\theta$), given that both $x < 1 - \theta$ and $y < 1 - \theta$, and $x + y = 1$?
So far I ...
3
votes
1answer
225 views
Finding tight constraints on a linear inequality
I have $a^\intercal M b > 0$, where $\forall a_i > 0$, $\forall b_j > 0$, and M is known. I'd like to find a tight linear constraint on $b$ which is independent of $a$ (other than the ...
3
votes
2answers
130 views
Calculate max/min of $x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$
What is a good way to calculate max/min of
$$x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$$
where $x_1+y_1+z_1+w_1=a$ and $x_2+y_2+z_2+w_2=b$ and $x, y, z, w, a, b \in \mathbb{N} \cup \{0 \}$, and please explain ...
1
vote
2answers
250 views
A question on inequality of arithmetic and geometric means
Let $x_i>0, i=1,...,n$ and $x_1+..+x_n=K$. From the inequality of arithmetic and geometric means, we have
$$x_1x_2...x_n\le \left( \frac{x_1+x_2..+x_n}{n} \right)^n$$
The equality holds if and only ...
1
vote
1answer
81 views
How can I find $\sup -\frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$ for $x_1 x_2 > 0$?
I need to find a constant $a$ such that for all $x_1 x_2 > 0$:
$$a > - \frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$$
that is to say the supremum of the term on the right hand side. My question is how to ...
2
votes
1answer
100 views
Linear complementarity problem - Classification
For Linear complementarity problems (LCP) like
$\mathbf{Mz}+\mathbf{q} \ge \mathbf{0}$
$\mathbf{z} \ge \mathbf{0}$
$\mathbf{z}^{\mathrm{T}}(\mathbf{Mz}+\mathbf{q}) = 0$
there exists a vast amount ...
-2
votes
1answer
260 views
Polynomial problem
From http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext2_05.pdf:
Suppose that $a$ and b are positive real numbers, and let $f(x)=\frac{a+b+x}{3(abx)^{\frac13}}$ for $x ...
2
votes
1answer
108 views
Inequality based on the minimum of a function
Given $1<q\leq2$ and $0\leq p\leq1$, let us consider the following function:
$$\phi\left(\alpha\right)=p\times\left|1-\alpha\right|^{q}+\left(1-p\right)\times\left|1+\alpha\right|^{q}$$
The ...
2
votes
0answers
205 views
Sharp (Reverse) Harmonic-Arithmetic Mean Bounds
Let $\mathbf{x} =$ {$x_{i}$} be a set of $n$ positive reals. In every good book on inequalities, one finds the classical result
\begin{eqnarray}
AM(\mathbf{x}) \geq GM(\mathbf{x}) \geq ...
