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I don't know what was meant in that exercise, but your revised conjecture is certainly true and well-known. Here is an elementary proof. The assumptions (local injectivity, continuity and segment-to-s …

The catch is in the word "canonical". If $V$ is positive, then $V^\perp$ is transverse to $V$ and hence naturally isomorphic to $L/V$ (by means of the projection $L\to L/V$ restricted to $V^\perp$). …

Update. This answer answers completely different question, see comments. Namely "positive" is substituted by "positive definite", norm is used instead of spectral radius, and quantifiers are different …

The sum is zero for all $k<p^2-1$. Assume that $k$ is a multiple of $p-1$ and $k<p^2-1$. Divide all matrices into classes of the form $\{A,A+1,A+2,\dots,A+(p-1)\}$ . Summimg the $k$th powers over su …

This is true and well known. By the minimax principle, $\alpha_k$ is the minimum over all $k$-dimensional subspaces of the norm of the quadratic form $v\mapsto(v,Yv)$ restricted to the subspace. And s …

Yes there are infinitely many such values of $n$ and the sequence satisfies the rule you observed. The proof is straightforward but technical. Let $x_1,\dots,x_n$ be the solution. Add $x_0=0$ and $x_ …

The image of the $L_1$-ball is the convex hull of the images $f(\pm e_i)$ of the basis vectors $e_1,\dots,e_n$ and their opposite ones. So you are given $n$ pairs of opposite points in $\mathbb R^m$ a …

No. Consider solid angles in $\mathbb R^2$. They all (except the half-plane) are isomorphic, yet one may be strictly included in another. Furthermore, a generic cone in $\mathbb R^d$, $d\ge 3$, has a …

To me continuity is more geometric and intuitive than the rest of the argument (which is purely algebraic manipulation). So I take the liberty to mis-read you question as follows: Is it possible to …

A homogeneous linear system does not have a nontrivial nonnegative solution if (and only if) some linear combination of the equaltions yields a nontrivial equation with nonnegative coefficients. Nothi …

A stronger fact holds: If $\lambda_1\le\lambda_2\le\dots\le\lambda_m$ are the eigenvalues of $A$, and $\mu_1\le\mu_2\le\dots\le\mu_p$ are eigenvalues of $Z^TAZ$, then $\mu_k\ge\lambda_k$ for all $k=1, …

Here is a simple proof that the property holds only for Euclidean norms, at least if the norm in question is $C^1$ smooth and strictly convex. Surely it was known way before Gromov was born. Let $S$ …

You are essentially asking whether a non-expanding linear map $A:(\pi,\|\cdot\|_\infty)\to(\mathbb R^3,\|\cdot\|_\infty)$ can be extended to a non-expanding linear map $B:(\mathbb R^3,\|\cdot\|_\infty …

The answer is infinity for $n>2$. Suppose that there is an $i$ such that $T^i(x)=0$ for all integer vectors $x$. Then the same follows for all rational vectors by homogenuity, and then for all real v …

There are 3 cases. Case 1. The field is finite. Then, as Charles Matthews pointed out, the primitive element theorem does the job. Case 2: The intersection of the subfields is infinite. This is cove …