All Questions
3
questions
5
votes
2
answers
169
views
Must a $𝐶^1$ curve with constant angular momentum alternate between a straight line or a circular arc?
Let $\alpha:(0,L) \to \mathbb{R}^2$ be a $C^1$ curve satisfying $|\dot \alpha|=1$, and assume that $\alpha(t) \times \dot \alpha(t)$ is constant.
Does one of the following hold?
$(1)$ $\alpha$ is ...
- 25.3k
4
votes
2
answers
84
views
A polygon with constant angular momentum bounds a circle
Let $\alpha:[0,L] \to \mathbb{R}^2$ be a piecewise affine map satisfying $\alpha(0)= \alpha(L)$ and $|\dot \alpha|=1$. Supopse that $\alpha(t) \times \dot \alpha(t)$ is constant.
How to prove that $\...
- 25.3k
2
votes
1
answer
205
views
Polygon of maximum area contained in compact, convex subset of $\Bbb{R}^2$?
Let $H$ be a compact, convex subset of $\Bbb{R}^2.$ For a given $m\ge 3,$ let $P_m$ be a polygon of maximum area which contained in $H$ and has atmost $m$ sides. Then $$\dfrac{Area(P_m)}{Area(H)}\ge\...
- 18.4k