# Tagged Questions

**5**

votes

**1**answer

152 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 ...

**2**

votes

**0**answers

146 views

### An isoperimetric type maximization problem with a barrier.

I'm trying to minmize a particular functional which depends on a curve with fixed endpoints which lies below a fixed line in $\mathbb{R}^2$. Here are the details:
Let $(r(\theta), \theta)$ be a ...

**4**

votes

**0**answers

248 views

### What are the most general types of curves in $\mathbb{R}^2$ for which Gauss-Bonnet holds?

I would like to know what is the most general form of the Gauss-Bonnet theorem in the plane for curves. It is well known for that for any piecewise $C^2$ simply connected curve with corners, one has
...

**2**

votes

**1**answer

164 views

### Is there a characterization of generalized constant mean curvature surfaces?

It is a well known result of Alexandrov that the only compact, connected, constant mean curvature surface is the ball.
There is a generalized notion of curvature known as generalized mean curvature ...

**4**

votes

**2**answers

468 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 ...

**3**

votes

**0**answers

302 views

### Generalization of First Variation of Area

The area of an $m$-rectifiable varifold in $n$-dimensional space can be expressed in terms of the surface divergence. More precisely, if $M$ is $m$-rectifiable, $\Omega$ is open, $\eta$ is a $C^1_c$ ...

**0**

votes

**1**answer

126 views

### optimize with respect to domain shape

Let $\Gamma$ be the set of all closed $C^2$ curves in the plane which enclose unit area and let $\Omega$ be the set of all subsets of $\mathbb{R}^2$ that are enclosed by some curve in $\Gamma$. Now ...

**12**

votes

**6**answers

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 ...