46 problems

Thom's gradient conjecture. If $\gamma(t)\to x_0\in\mathcal{U}$, then $\gamma$ has a tangent at $x_0$; equivalently, the limit of secants

Let $\mu_1 \in \mathcal{D}_+(S^1)$ be a finitely supported, non-negative measure. For the unit ball $\mathcal{B}_1(\mu_1)$ and its extreme points…

Let $L$ be a convex function on ${\mathbb R}^n$, let $L'(x,\theta)$ denote its directional derivative at $x$ in the direction $\theta$, and define … where $\sigma_{n-1}$ is the uni…

The Vaĭnberg–Brègman setting concerns left $D_\Psi$-projections and sunny quasinonexpansive right $D_\Psi$-retractions, with the Euler–Legendre transform relating the corresponding…

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 $\Omega_X$ be the domain of a convex function $u:\Omega_X\to\mathbb{R}$, and let $v$ be its convex conjugate. Local detachment conjecture. The gradient $\nabla u$ is $L$-Lipsch…

Let $(\mathbb{R}^N)^n$ be the phase space associated with a language-model geometrization, and let … be a differentiable extension of the language-model entropy function. The entro…

Let $E\subset\mathbb{R}^n$ be compact, let $f:E\to\mathbb{R}$, and let $M>0$ and $\eta\geq 0$. Assume that there are constants $k^\#\in\mathbb{N}$ and $C_1>0$ depending only on $n$…

Coincidence conjecture. Under the density assumption on $\mathcal{G}_0$, the approximate irreducible components defined using $\mathcal{G}_0$ and those defined in the alternative c…

Let $\Omega$ be the underlying space, let $\mu$ and $\nu$ be the source and target measures, and let $\Gamma_{\mathcal{F}}(\mu,\nu)$ denote the set of $\mathcal{F}$-transports betw…

Let $X$ be a topological vector space and let $C\subseteq X$ be a convex set. The quasi-relative interior of $C$, denoted $qri C$, and the face relative interior of $C$, denoted…

KPZ shape-function conjecture. The shape functions are strictly convex and differentiable; equivalently, they have no flat edges and no corners.

Carathéodory-number conjecture. For a function from $L$ to be an optimal approximation in uniform norm, it is enough that it is optimal at $c(\mathcal{C}_{H_L})+1$ extreme points.…

Geometry conjecture. There exist $v_1,\dots,v_m\in H$ such that

Let $C$ be a polyhedral ordering cone, and let $\beta_1$ and $\beta_2$ denote the constants associated with the distance comparisons defined earlier in the paper. Finiteness conjec…

Let $C$ be an ordering cone in $\mathbb{R}^D$, and let $\beta_1$ and $\beta_2$ denote the constants associated with the distance comparisons defined earlier in the paper. Assume th…

Let $a(\omega)$ be an optimal policy with support $\Xi$, and define … The conditions in Proposition 1.4 are that, when $\Xi$ is $X$-maximal for some set $X$ and $\Xi\subseteq X$, o…

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…

Legendre-transform adjunction conjecture. In the Bregmanian setting, under a suitable choice of these categories, the Legendre transform is an adjunction, with the equivalence of c…

Let $Q(c)$ denote the set of reserve vectors having marginal price $c$ for a strictly convex trading function. In the Curve example, scaling the reserves by a nonzero constant chan…

Conjugate random-set optimality conjecture. Compared to the classic results, one may conjecture that $\mathcal{X}_B(\lambda \xi)$ is the optimal solution to Problem. The conjecture…

The geometry conjecture. There exist vectors $v_1,\ldots,v_m$ in $X$ such that

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…