6 problems

Let $\mathcal{K}(\mathbb{S}^n)$ denote the set of spherically convex bodies, and let $\Omega_s(K)$ be the floating area of $K\in\mathcal{K}(\mathbb{S}^n)$: … For each $K$, let…

Generalized-curvature extension conjecture. It is generally conjectured that Theorem Gruber's best-approximation result holds for arbitrary convex bodies when the classical Gauß-Kr…

Stancu's algebraic-generation conjecture. The set of $p$-affine surface areas, together with algebraic operations, can generate, by taking its closure, all integrals of the form

Stancu's generation conjecture. Any $SL(n)$ invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized $p$-af…

Let ${\cal K}_0^n$ be the space of convex bodies in $\mathbb R^n$ containing the origin in their interiors. A valuation $\Psi:{\cal K}_0^n\to(-\infty,\infty]$ is homogeneous of deg…

Let ${\cal K}_0^n$ be the space of convex bodies in $\mathbb R^n$ containing the origin in their interiors. A valuation $\Psi:{\cal K}_0^n\to(-\infty,\infty]$ is lower semicontinuo…