Timeline for An isoperimetric inequality for curve in the plane?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 7, 2019 at 13:41 | comment | added | Ben McKay | It generalizes to $\int_{\partial U} r \ge n \operatorname{Vol} U$, for $U \subset \mathbb{R}^n$ a bounded domain with $C^1$ boundary, with the origin in its interior, by the same proof as Piotr Hajlasz's. Maybe even with the origin on the boundary. | |
| Mar 6, 2019 at 19:03 | comment | added | Fedor Petrov | If R is convex and origin is inside (or even if it is star-shaped with respect to the origin), the triangles cover it without overlaps. Otherwise overlaps are possible. But definitely any point p in R is covered by the segment 0s, where s is the point in which the continuation of the ray 0p meets the boundary of R. | |
| Mar 6, 2019 at 17:25 | comment | added | T. Amdeberhan | @FedorPetrov: I almost agree, but does this not matter whether the curve is convex or concave? In other words, is the inequality local or an average? | |
| Mar 6, 2019 at 13:16 | comment | added | Fedor Petrov | Looking infinitesimally, $\frac12r\Delta s$ is not less than the area of a triangle with vertex at origin and side $\Delta s$. Such triangles cover $R$, thus the inequality. Equality takes place only if the radius-vector is always orthogonal to the tangent line, that means that the derivative of $r$ is zero. | |
| Mar 6, 2019 at 6:06 | comment | added | T. Amdeberhan | Good idea there. | |
| Mar 6, 2019 at 1:18 | vote | accept | T. Amdeberhan | ||
| Mar 6, 2019 at 1:06 | history | edited | Piotr Hajlasz | CC BY-SA 4.0 |
I added: centered at the origin.
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| Mar 6, 2019 at 0:21 | answer | added | Piotr Hajlasz | timeline score: 7 | |
| Mar 5, 2019 at 23:48 | comment | added | RBega2 | If $C$ is regular enough (e.g. $C^1$) doesn't this this follow from the diveregence theorem applied to $(x,y)$ and the Cauchy-Schwarz inequality? | |
| Mar 5, 2019 at 23:29 | history | edited | YCor | CC BY-SA 4.0 |
added more context in title
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| Mar 5, 2019 at 23:24 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |