All Questions
Tagged with convexity open-problems
8 questions
94
votes
2
answers
6k
views
Volumes of sets of constant width in high dimensions
Background
The $n$-dimensional Euclidean ball of radius $1/2$ has width $1$ in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between ...
- 24.7k
38
votes
0
answers
1k
views
Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
- 7,194
16
votes
0
answers
2k
views
An open problem in convex geometry
Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
- 13.5k
10
votes
1
answer
539
views
Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?
In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$.
Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are
...
- 7,024
10
votes
0
answers
265
views
Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
- 7,194
8
votes
1
answer
620
views
Small quadrilaterals containing a given convex region
It is easy to prove that
(*) Every convex planar set of area 1 is contained in a quadrilateral of area 2.
It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
- 7,276
3
votes
0
answers
95
views
Effective radius of section of a convex set compared to that of the convex itself
The effective radius $er(A)$ of a $n$-solid $A$, is defined by Schramm (see the question by Gil Kalai
Volumes of Sets of Constant Width in High Dimensions)
to be the radius of the $n$-ball that has ...
- 469
2
votes
0
answers
155
views
Inscribed square and convexity
Let $b(X)$ be the boundary of any $X$ subset of the plane.
Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...
- 469