12 problems

Let $\mathcal{J}^{1}$ denote the class of smooth Jordan curves, and let $\mathcal{Q}$ denote the specified circular quadrilateral. Makeev's conjecture. The class $\mathcal{J}^{1}$…

Let $\gamma:S^1\to\mathbb{RP}^n$ be a convex curve, and consider any $(k+1)(n-k)$ pairwise distinct projective $k$-dimensional osculating subspaces to $\gamma$. A configuration is…

Let $\gamma:S^1\to\mathbb{RP}^n$ be a smooth curve. For $k=0,\ldots,n-1$, let $D_k(\gamma)$ denote the union of the $k$-dimensional osculating subspaces to $\gamma$, and let its de…

Let $N>0$, let $I$ be the interval under consideration, and let $f:[0,N]\to[0,N]$ be a strictly convex function such that $f\in C^3([0,N])$ and $f^{(3)}(x)\neq 0$ for all…

Let $\gamma:\mathbb{R}\to\mathbb{R}_{m-1}[u]$ be a convex curve, meaning that any $m$ distinct points along $\gamma$ are linearly independent. Let $z_1,\dots,z_{d(m-d)}$ be distinc…

Let $\Gamma:(x,f(x))$ be a strictly convex $C^3$ graph, with $f”$ weakly monotonic and vanishing at most once, and let $N$ be a positive integer. Write…

Benguria–Loss conjecture. The following inequality holds:

Let $\mathcal{L}\mathbb S^3(\varepsilon_1,\varepsilon_2)$ denote the indicated spaces of locally convex curves on the 3-sphere, where each…

Let $\mathcal{L}\mathbb S^3(\mathbf{1},-\mathbf{1})$ be the space of locally convex curves on the 3-sphere with the indicated endpoint data, let $L$ denote the associated map to cu…

Guggenheimer's conjecture. The minimal volume product $V(C)V(C^*)$ is attained when $C$ consists of the longest diagonals of a regular $(2k+1)$-gon with centre $0$, taken always in…

Two-tripod conjecture. Every smooth closed convex curve in the spherical or hyperbolic geometry has at least two tripod configurations.

DNA inequality conjecture. The DNA inequality holds for arbitrarily small dents of $P$ along side $AB$ if and only if