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Surface area and volume uniquely determine a shape 
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#1
May914, 07:49 AM

P: 56

Is this so? I cannot think of a counterexample and it is too general a statement to prove.



#2
May914, 08:00 AM

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P: 3,458

A counterexample: two human hands, identical except that on is a left hand and the other is a right hand.
There are many more: a cube with two cylindrical protrusions, has the same area and volume no matter how you move the protusions around. 


#3
May914, 08:09 AM

P: 56

Are there smooth manifolds (excepting mirroring)?
Basically you cut a shape in two parts and glue theme on another one. Generalising your construction: construct a shape with complementary protrusions (sort of a hermaphroditic shape), cut another shape along the protrusion plane and fit the two parts in the respective protrusions. It seems to me that there will always be an edge. 


#4
May914, 08:49 AM

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P: 3,458

Surface area and volume uniquely determine a shape



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