9 problems

Characterization conjecture. The operator $A$ must either be the subdifferential $\partial f$ of a proper convex lower semicontinuous function $f$ of the form described in the pape…

Let $X$ be the domain of a Lipschitzian function $F:X\to\mathbb{R}$, and let $\nabla F(\boldsymbol{x})$ denote the set of subgradients of $F$ at $\boldsymbol{x}$. For…

Let $r$ be the rational function defined in the paper, let $P$ be a polytope, let $f(x)=r(x)+I_P(x)$, and let $v$ be a vertex of $P$ satisfying $\xi_1(v)=\xi_2(v)=0$. Vertex conjec…

Let $C\subset\mathbb{R}^k$ be a compact convex domain and let $f:C\to\mathbb{R}$ be bounded and measurable. Write $f^{\ast n}$ for the $n$-fold sup-convolution of $f$, let…

Let $\mathfrak{H}_{+,m}$ denote the cone of positive semidefinite Hermitian $m\times m$ matrices, let $\mathbb T^d$ be the $d$-dimensional torus, and let $\mathrm{d}M$ be a nonnega…

Let $\Omega \subset \mathbb{R}^n$ be a compact convex polytope, and let $\Gamma_k$ denote its $k$-skeleton. A convex function is a function whose epigraph is convex. Higher-dimensi…

Second-derivative characterization. The function $f$ is $h$-convex if and only if

Equal minimal subgradient norm conjecture. If

Convexity and reciprocal-concavity conjecture. For each $N\in\mathbb{N}$, the function $p_N$ is convex; indeed, $1/p_N$ is concave. This claim concerns the network objective functi…