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Results tagged with calculus-of-variations
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user 6094
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
25
votes
Accepted
How can you compute the maximum volume of an envelope(used to enclose a letter)?
Your question is a variant of the teabag problem.
I don't believe an exact answer is known, but
for the $1 \times 1$ square teabag, the maximum volume is about $0.2$:
(Image …
- 151k
5
votes
Area of the minimal surface of a non-planar quadrilateral in 3d
This paper seems to give a partial answer to the posed question, for
skew quadrilaterals that project to rectangles:
Furui, Sadataka, and Bilal Masud. "Numerical calculation of a minimal surface u …
- 151k
7
votes
Largest possible volume of the convex hull of a curve of unit length
Here is an image of the optimal open convex curve.
Taken from Open Problems from CCCG 2012,
based on this paper, which cites Nudel'man (1975):
Paolo Tilli.
"Isoperimetric inequalities for conve …
- 151k
4
votes
Smallest area shape that covers all unit length curve
P.A.P. Moran proved in 1946, in "On a Problem of S. Ulam" [J. London Math. Soc. 1946 s1-21: 175-179] this theorem:
If $C$ is a curve of unit length in the plane, and $|K$| is the area of its small …
- 151k
3
votes
Accepted
Names of certain surfaces
If Surface I yet has no name, I would christen it Winged Victory. :-)
- 151k
6
votes
Closed curve whose neighborhood is as large as possible
Just to emphasize Thomas Richard's remark about smoothness, unless I've miscalculated, a $\frac{1}{4} L$-square leads to area
$$2 \epsilon L - \epsilon^2 (4-\pi) < 2 \epsilon L \;.$$
Added …
- 151k