12 problems

Let $d\ge 1$. Suppose $D(x)$ has finite support $\mathcal{U}$. If $D(x)$ is uniformly distributed on $\mathcal{U}$, and if every $x\in \mathcal{U}$ is on the boundary of…

Let $1<p<\infty$ and $1\leq k\leq n$. For a matrix $A$ and matrix subspace ${\mathcal M}$, let $R^{(p,k)}=A-B^{(p,k)}$ range over residuals of best approximations with respect to t…

Let ${\mathcal M}$ be a matrix subspace, let $A$ be a matrix, and let $Y_p$ be the unique best approximation to $A$ in ${\mathcal M}$ with respect to the $c_p$ norm for…

Local-global norm conjecture. The local-global principle for norms holds in non-cyclic extensions of number fields.

Let $N$-qubit states be represented by vectors $\rho$, and let the minimum $l_p$ ball containing all separable states have radius … for $1\leq p\leq2$. For $p\geq2$, the source sta…

Minimal separable-ball conjecture. Every $l_p$ ball in Theorem 9 is the smallest $l_p$ ball that contains all $N$-qubit states. The proof of Theorem 9 is given in the paper, but th…

Norm-extension conjecture. Then $\lVert\cdot\rVert$ is the restriction to $G$ of a norm on $\mathbb Z^3$.

Let $\mathscr{C}(\{p,\kappa_{d,p}\})$ denote the class of alternatives against which the $p$-norm based test with critical values $\kappa_{d,p}$ is consistent, and choose all criti…

Convex-optimization tractability conjecture. Methods based on convex optimization could solve this problem in polynomial, but not strongly polynomial, running time.

Bartels's conjecture. The Hasse norm principle should hold for every $k$-adequate extension.

Let $D$ be a positive nonsquare integer, let $\mathbb Z[\sqrt D]$ be the quadratic order, and let $N$ be the minimum of the absolute values of negative norms of elements of…

Let $f$ be a nonconstant polynomial whose complex coefficients have unit magnitude, and let $\|f\|_2$ and $\|f\|_\infty$ denote its $L^2$ and $L^\infty$ norms on the complex unit c…