14 problems

Meissner's conjecture. Meissner's tetrahedra minimize volume among all three-dimensional convex bodies of constant width $d$.

Reuleaux triangle maximality conjecture. The Reuleaux triangle maximizes the first eigenvalue of the $p$-Laplacian among all bodies of the same constant width.

Approximation conjecture. Any spherical convex body of constant width $\pi/2$ can be approximated by a sequence of spherical convex polytopes of constant width $\pi/2$.

Let $K$ be a convex body of constant width in ${\mathbb{R}}^4$, and let $C$ be a regular cross polytope, also called a generalized octahedron. Makeev's conjecture. Every constant-w…

Let $K$ be a convex body of constant width in ${\mathbb{R}}^n$, and let $D_n$ denote the dual of the difference body of a regular simplex. Makeev's conjecture. Every constant-width…

Let $\mathbb{S}^n$ be the unit sphere in $\mathbb{R}^{n+1}$, with spherical $n$-measure $\mu_n$. Let $\mathcal C_n(w)$ denote the class of spherical convex bodies of constant width…

Blaschke–Lebesgue conjecture in dimension three. The Meissner bodies are the unique minimizers of volume among bodies of constant width $1$ in dimension $3$.

Let $K\subset\mathbb{R}^3$ be a constant width body. For almost every outward unit normal, let $r_{\text{min}}$ and $r_{\text{max}}$ denote its minimum and maximum principal radii…

Three-dimensional Blaschke–Lebesgue conjecture. The three-dimensional analogue of the Reuleaux triangle is the constant-width shape minimizing the volume.

Let $d\geq 3$, let $c_{BL}(r,d)$ denote the minimum spherical volume of a wide $r$-ball-body in the sphere … and let ${\Delta}_d(\frac{\pi}{2})$ be the spherical regular simplex of…

Let $\mathbf{K}$ be a three-dimensional convex body of constant width. Constant-width illumination conjecture. The illumination number satisfies $I(\mathbf{K})=4$. The known genera…

Translative constant-width conjecture. If $d\geq3$ and some $K\in\mathcal{K}_d$ satisfies the translative equal volume property, then $K$ is a convex body of constant width in a Eu…

Translative constant volume conjecture. Then $K$ is an ellipsoid.

Let $d\ge 4$ and consider convex bodies of constant width $\frac{\pi}{2}$ in the unit sphere $\mathbb{S}^{d-1}$. A $(d-1)$-dimensional regular simplex of edge length…