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2

Using (say) the Jordan form, approximate $A$ by a nonsingular matrix $C_t$ so that $C_t\to A$ as $t\to0$. Then $$C_t^\top (C_t X^{-1} C_t^\top + B)^{-1} C_t\le C_t^\top (C_t X^{-1} C_t^\top)^{-1} C_t=X.$$ Letting now $t\to0$, we get $$A^\top (A X^{-1} A^\top + B)^{-1} A \le X,$$ as desired.

14

Yes, because the OP stated that the ground field is $\mathbb{R}$, one can simply take the octic polynomial $$Q(v_1,v_2,\ldots,v_n) = \sum_{1\le i < j\le n} \bigl((v_i,v_i)(v_j,v_j)-(v_i,v_j)^2\bigr)^2,$$ which will do the trick.

4

An affirmative answer to your question would follow from a famous conjecture about envy-free allocations. It is also known that if you replace the '2' in $b$-goodness with '$\frac 12$', then it is true. A good paper to read (first) on the topic is Almost Envy-Freeness with General Valuations. An allocation of some items among some players is envy-free up to ...

0

Just for those who might want to know, I think the min-max theorem mentioned by @Michael Renardy is Courant-Fischer: Supporse a real symmetric matrix A's eigen values are $\lambda_1 \leq \lambda_2 \leq ... \leq \lambda_n$. then k-th eigen value: $$\lambda_k = \underset{V_k}{\text{min}} \space max\{x^TAx | x \in V_k, ||x||_2 = 1\}$$ $V_k$ is any k-...

3

I was recently thinking about this question again and I came up with another way to explain the result $$\mathcal H_{\lambda}=\left\{ h(i,j)\right\}_{(i,j)\in \lambda}\succcurlyeq \left\{ h^{*}(i,j)\right\}_{(i,j)\in \lambda}=\mathcal H^{*}_{\lambda} \tag{1}$$ in a way that (i) combinatorially describes the sequence of Robin Hood moves to get from $\mathcal ... 0 Regarding the inequality for$\Vert AB \Vert_2 $, although it does not hold in the form you wrote it as Denis pointed out, replacing the 2-norm by the 1-norm leads to Holder's inequality for Schatten norms: \begin{equation} \Vert AB \Vert_1 \leq \Vert A \Vert _p \Vert B \Vert _q \end{equation} 10 The inequality is false for$a=1799, b=105, c=1024, d=4$. This counterexample was found and verified with Mathematica, as follows: 1 Weyl's inequality, and Hua's inequality. They are quite important from the viewpoint of analytic number theory. 5 I have nothing to add to the Fourier-type approach suggested in the question, but for those curious, thought it useful to outline the combinatorial solution to the problem that I know (I believe this is the same as the IMO official solution, and claim no originality). One thing to add is that, although induction is a crucial part of the proof, we do not use ... 0 This can be done with help of Maple in such a way. First, we find the estimated expression explicitly by a := (exp(x)*int(sin(exp(t)), t = x .. x + 1) assuming x::real;$ {{\rm e}^{x}} \left( -{\it Si} \left( {{\rm e}^{x}} \right) +{\it Si} \left( {{\rm e}^{x+1}} \right) \right) $In fact, the integral is reduced to another integrals. Next, the ... 2 We have$0\le a\le b\le1$and$T\in(0,\infty)$. We want to know when there is a positive constant$C$such that $$\int_0^T dt\, \int_a^b dx\, u^2(x-t)\geq C\int_0^1 dx\,u^2(x) \tag{1}$$ for all measurable functions$u\colon\mathbb R\to\mathbb R$such that$u(x)=0$for$x\notin(a,b)$. The answer is: never. Indeed, without loss of generality$a<b$. The left-... 6 For any unit vector$x\in L^2(\mathbb{R}^n)$, $$\|\sqrt{A+B}\,Cx\|^2=(\sqrt{A+B}\,Cx,\sqrt{A+B}\,Cx) =(ACx,Cx)+(BCx,Cx)=\|\sqrt A\,Cx\|^2+\|\sqrt B\,Cx\|^2 \le(\|\sqrt A\,Cx\|^2+\|\sqrt B\,Cx\|)^2 \le(\|\sqrt A\,C\|+\|\sqrt B\,C\|)^2.$$ So, $$\|\sqrt{A+B}\,C\|\le\|\sqrt A\,C\|+\|\sqrt B\,C\|.$$ 8 $$\|\sqrt{A+B}Cx\|^2=(\sqrt{A+B}Cx,\sqrt{A+B}Cx)=\\ ((A+B)Cx,Cx)=(ACx,Cx)+(BCx,Cx)=\|\sqrt{A}Cx\|^2+\|\sqrt{B}Cx\|^2,$$ taking the supremum over unit vectors$x$we get $$\|\sqrt{A+B}C\|^2\leqslant \|\sqrt{A}C\|^2+\|\sqrt{B}C\|^2\leqslant (\|\sqrt{A}C\|+\|\sqrt{B}C\|)^2.$$ 4 For any real numbers$u,v,c$such that$u\le c\le v$, let$\mu_{c;u,v}$denote the unique probability distribution on the set$\{u,v\}$with mean$c$. Your generalization of Jensen's inequality follows immediately from the well-known fact that any probability distribution$\mu$on$\mathbb R$with a given mean$c\in\mathbb R$is a mixture of probability ... 0$C=1.0986702957852257\$ is the solution. It can be proved by using the Langrange multipliers.

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