3
votes
1answer
192 views

Delta-convex functions and inner products

A delta-convex (d.c.) function is one which can be written as the difference of two convex functions. The space of d.c. functions includes all C2 functions, and is interesting because it allows many ...
3
votes
1answer
189 views

Riemannian metric on a space of “not-quite-smooth” (hyper)surfaces?

Apologies if the title's a bit vague; I'll try to explain everything below, and please let me know if I should clarify anything. I've been looking for a while at variational problems on polytopes. ...
1
vote
1answer
456 views

Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...
3
votes
1answer
184 views

Integral of a quadratic on a polygon (variations of discrete surfaces)

This question is pretty simple to state: given two linear functions on a polygon, I'm looking for a formula for the integral of their product which depends only on the values at the (unlabelled) edges ...
2
votes
0answers
297 views

Partial feedback linearization (Control theory)

Greetings, I'm trying to understand a theorem about partial feedback linearization from a paper "On the largest feedback linearizable subsystem" by R.Marino (you can find it here: ...
6
votes
1answer
263 views

Constructing a hypersurface with given outer normals

Pick a point on each of the positive half-axes in $\mathbb{R}^n$. Put a (unit-norm?) vector at each of the n points. (a) Is there a hypersurface in the orthant $\mathbb{R}^n_+$ going through these n ...
5
votes
2answers
414 views

Maximize the intersection of a n-dimensional sphere and an ellipsoid.

I have the conjecture that the volume of the intersection between an $n$-dim sphere (of radius $r$) and an ellipsoid (with one semi-axis larger than $r$) is maximized when the two are concentric, but ...
2
votes
2answers
544 views

Mean curvature of polyhedral surfaces

Whilst working on this problem, I've come across the work of Konrad Polthier on generalizing the notion of curature to discrete surfaces; much of his work is covered in his thesis, "Polyhedral ...
20
votes
2answers
1k views

Functions whose gradient-descent paths are geodesics

Let $f(x,y)$ define a surface $S$ in $\mathbb{R}^3$ with a unique local minimum at $b \in S$. Suppose gradient descent from any start point $a \in S$ follows a geodesic on $S$ from $a$ to $b$. (Q1.) ...
9
votes
2answers
2k views

An optimization problem for points on the sphere (master's dissertation)

First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
21
votes
2answers
1k views

What is the best way to peel fruit?

A mango made me wonder about this. (See also this question, which is in a similar spirit.) Fix $L >0$ and a smooth body (possibly nonconvex—pears or bananas are fair game!) $B \subset ...