2
votes
1answer
104 views

Distance from constant width bodies

EDIT As @David has observed, my conjecture was clearly wrong for $\ n:=2.\ $ Let me still give it a chance for $\ n\ge 3$. I'll call a family $\ F\ $ of bound closed convex subsets of $\ \mathbb ...
2
votes
1answer
126 views

intersection of the unit cube and a hyperplane containing the main diagonal

Let $A$ be a linear $m$-dimensional subspace of $\mathbf{R}^n$ $m < n$, containing the point $(1,1,\ldots,1) \in \mathbf{R}^n$, and consider the intersection of $A$ and the unit cube $\Delta_n$ ...
7
votes
0answers
101 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
8
votes
1answer
222 views

Largest convex hull of a unit length path

What is the largest area possible for the convex hull of a path of unit length lying on a plane? For what paths is that largest area attained?
7
votes
2answers
358 views

Two cubes in unit cube

A cube of side one contains two cubes of sides a and b having non-overlapping interiors. How to prove the inequality $a+b \le 1$? The same question in higher dimensions. It was asked, but not answered ...
0
votes
1answer
338 views

On the number of lines of given points

Hi all, I have a question Concerning Beck's theorem. I have read it from http://en.wikipedia.org/wiki/Beck%27s_theorem and I have two questions : I suppose Beck's theorem doesn't hold when instead ...
22
votes
3answers
646 views

Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
12
votes
2answers
220 views

Smallest containing simplex

Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$. What is known about $V_n$? Is there a ...