Tagged Questions

298 views

Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of volume vol$(K) = V$ inside the unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$. You are permitted to probe with a (one-dimensional) ...
571 views

Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, ...
370 views

A toy model for the t-section problem

Let $S(x)$ be the area of the yellow curvilinear triangle. I'd like to find a graph for which $S(x)=H(x)$ where $H$ is some prescribed function (small, smooth, vanishing near the endpoints to any ...
2k views

The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself. Suppose we are given $n$ ...
696 views

Dissecting trapezoids into triangles of equal area

[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark] The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...
869 views

Diameter of universal cover

Let $M$ be Riemannian manifold and $\tilde M$ be its universal cover (with induced metric). What is the upper bound for $k=\mathop{diam}\tilde M/\mathop{diam} M$ in terms of $m=|\pi_1(M)|$ (or ...
571 views

What is known about polyhedra nets that allow overlapping?

It is an open problem that the net of any convex polyhedron can be unfolded onto a flat plane with no overlapping. Is anything known if we allow x faces to overlap? For example, is it known if any ...
2k views

Convex hull in CAT(0)

Let $X$ be complete $\mathop{CAT}(0)$-space and $K\subset X$ be a compact subset. Is it true that convex hull of $K$ is compact? Comments: Convex hull of $K$ = intersection of all closed convex ...
650 views

Milnor's cartography problem

Let $\Omega$ be a round disc of radius $\alpha<\pi/2$ on the standard sphere. It is easy to construct a $(1,\tfrac{\alpha}{\sin\alpha})$-bi-Lipschitz map from $\Omega$ to the plane. Is it true ...
Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space and $r$ is the distance from $\Sigma$ to the center of the ball. Is it true that \mathop{\rm area} \Sigma\ge ...