Search Results

In your specific case, $W(S_1,S_2)$ is the polydisk of radius $\frac12$. For $$|b\bar a|^2+|b\bar c|^2=|b|^2(1-|b|^2)\le\frac14$$ and conversely if $|w|^2+|z|^2\le\frac14$, one takes $b=\sqrt t\in(0,1 …

Here is a complete proof. One proves easily that if $S\in{\mathcal S}^n_{++}$ and if $zz^T\prec S$ (in the order between symmetric matrices), then $$\frac12 y^TX^{-1}y\ge z\cdot y-\frac12 {\rm Tr}(SX …

Yes Consider first the case where $f\in{\cal C}^2$. Then $$\nabla f(y)-\nabla f(x)=\int_0^1{\rm D}^2f(x+t(y-x))\cdot(y-x)\,dt.$$ There follows $$\|\nabla f(y)-\nabla f(x)\|\le\|y-x\|\int_0^1\|{\rm D}^ …

If the assumption is $(\psi''')^2\le\frac12\,\psi''\psi^{(4)}$, then it can be written as $(1/\psi'')''\le0$. Therefore $1/\phi''$ is concave. Since it is positive over $(0,+\infty)$, it must be non-d …

Not an answer (after all Fedja gave a satisfactory answer), but another approach of this problem, in terms of PDEs. Suppose that a continuous map $A:\bar\Omega\to{\bf Sym}_n^+$ satisfy the following p …

Write $c_k=p_1+\cdots+p_k$ with $p_k=c_k-c_{k-1}\ge0$. Then $$\sum_ic_ib_i-\sum_ic_ia_i=\sum_kp_k(s_k(b)-s_k(a))$$ with $s_k(a)=a_k+\cdots+a_n$. By assumption, $s_k(b)-s_k(a)\ge0$ and you are gone. …

The answer is Yes when $n=2$,but No when $n\ge3$. Here is the analysis. The differential $L_A$ of $A\mapsto A^{-1}$ is $L_A=-A^{-1}BA^{-1}$. Likewise, the Hessian is $$H_A[B]=2A^{-1}BA^{-1}BA^{-1}=\f …

The second derivative of a convex function, in the distributional sense, is a non-negative bounded measure. And conversely. If this measure $\mu$ contains a sum $\sum_na_n\delta_{x=x_n}$, where $(x_n) …

The Krein-Milman Theorem is used in the proof of Birkhoff's Theorem that the set of bistochastic matrices is the convex envelop of permutations matrices.

Are you really interested in the convex cone, or in the convex envelop ? The latter is easily determined by taking the intersection of the half-spaces containing all the pairs $(hh^T,h)$. Their equati …

Here is an elementary proof that the answer is positive, at least if you relax the condition that the extension be $C^2$. Lemma. Given $x_0\in C_M$, there exists a linear form $\lambda$ such that …