4 problems

Let $K$ be a convex body in a normed space, let $x\in\operatorname{int}K$, and let $B_n(K,x)$ denote the Bernstein factor at $x$ for polynomials of degree at most $n$. Write $w(K)$…

Let $(M,g)$ be a compact Riemannian manifold, and let … where each $\varphi_{\lambda_k}$ satisfies … For sufficiently small $r>0$, let $B_g(x,r)$ denote the geodesic ball centered…

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$, and let $\varphi_{\lambda}\in C^{\infty}(M)$ satisfy … For sufficiently small $r>0$, write $B_g(x,r)$ for the geodesi…

Asymptotic sharpness conjecture. For $1\leq p\leq\infty$, the constant $(1+r)^{1/p}$ in the estimate