6
votes
0answers
131 views

Choquet theory and Hilbert's fourth problem

The following text is an attempt to see Hilbert's fourth problem in a new light. Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line ...
4
votes
0answers
217 views

Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the variational problem $$ ...
5
votes
1answer
149 views

Minimizing the perimeter around an obstacle

Let $A\subset \mathbb{R}^n$ be a (measurable) bounded set, and consider the following optimization problem: minimize $P(X)$, the perimeter of a set $X$, where $X$ ranges over all Caccioppoli subsets ...
3
votes
0answers
167 views

Methods for generating metrics and minimizing variational dynamics of particles (masses or charges) on n-dimensional smooth manifolds

I am attempting to investigate transformations between two distinct sets of vertices on n-dimensional manifolds with a minimal change in the fundamental shape of the vertices. I will give some ...
8
votes
1answer
473 views

Maximal tetrahedra inscribed in ellipsoid

Pietro Majer quoted the theorem of Michel Chasles in his MO question, "Convex curves with many inscribed triangles maximizing perimeter," which states that the triangles of maximum perimeter inscribed ...
3
votes
1answer
177 views

A minimum problem of the CoV

I have the following minimum problem: $$\tag{1} \min_{u\in W^{1,1}(0,B)} \int_0^B (\sqrt{1+|u^\prime (t)|^2} -a)\ u^{k-1} (t) t^{h-1}\ \text{d} t $$ (where $B>0$, $0 < a < 1$, $h,k\in ...
4
votes
2answers
459 views

Minimal surface which divides a convex body into two regions of equal volume

Question. Given a convex body $\Omega$, what is the shape of a surface $\Gamma$ of minimal area which divides $\Omega$ into two regions of equal volume? Background/motivation. A 2D version of ...
9
votes
6answers
1k views

Smallest area shape that covers all unit length curve

On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve. What are the bounds of the shape's area if this ...