13 problems

Ratio-integrability continuity conjecture. Then $f$ has a continuous representative. This is the explicit formulation of the proposed stronger continuity statement for generalized…

Let $\Omega\subset\mathbb{R}^n$ be a domain, let $1\leq p\leq\infty$, and let $p^*$ denote the Hölder conjugate of $p$. Consider $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$…

Let $\Omega\subset\mathbb{R}^n$ be a domain, let $1\leq p\leq\infty$, and let $p^*$ denote the Hölder conjugate of $p$. Consider a Sobolev mapping…

Continuity conjecture for $C_B^{\infty}$. The analogue of the continuity condition in Proposition B should hold for the function $C_B^{\infty}$.

Conjecture. Condition … . The conjecture proposes that the stated continuity condition is not only sufficient but also necessary for convergence of the mutual information, independ…

Continuity conjecture. If the drift is weak at $x_0$, then $V_\mu(x_0,\cdot)$ is continuous at any point of $M$.

For $\alpha>0$, let … Here $c_{\alpha}$ is the asymptotic expected value associated with the $\alpha$-percentile rule. Gaither–Ward continuity conjecture. The function…

Strictness conjecture. All of the above implications are strict, meaning that none of them can be reversed. The conjecture is motivated by the differing strengths of the associated…

Let $X$ and $Y$ be metric spaces. A subset $L$ of $X$ is a compact image if there are a compact metric space $K$ and a uniformly continuous function $f\colon K\to X$ such that…

Bijectivity conjecture. This difficulty is intimately connected with the absence of a bijective property.

Positive-measure discontinuity conjecture. The set of discontinuities of $\check{\varrho}$ has positive Lebesgue measure. This is presented as a weaker version of the resist-impuri…

Connected-set continuity conjecture. Condition (2) characterizes continuity: $P$ is continuous if and only if $P(C)$ is connected for every connected subset $C$ of $E$.

Conjecture 2. If $F_n\Rightarrow F$, then