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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.
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 …
- 149k
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 …
- 149k
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 …
- 149k
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 …
- 149k
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 …
- 149k
3
votes
Accepted
Names of certain surfaces
If Surface I yet has no name, I would christen it Winged Victory. :-)
- 149k