These four inequalities define a hollow (zero volume) cube. Is it possible to describe a hollow cube

Simone Werner

Simone Werner

Answered question

2022-05-28

These four inequalities define a hollow (zero volume) cube. Is it possible to describe a hollow cube using a smaller system of (real scalar generalized-polynomial) inequalities?
x 2 1 y 2 1 z 2 1 ( x 2 1 ) 2 ( y 2 1 ) 2 ( z 2 1 ) 2 0

Answer & Explanation

aqueritztv

aqueritztv

Beginner2022-05-29Added 10 answers

Certainly there exist infinitely many possibilities of a single scalar equation that will do the job, using any function that is zero on the boundary of the unit cube and nonzero elsewhere. Here is one explicit example:
max ( | x | , | y | , | z | ) 1 = 0
If you prefer inequalities, you can just square the left-hand side and set it 0.
But is this just a mathematical version of code golf, or is there some other reason for this question?
Monfredo0n

Monfredo0n

Beginner2022-05-30Added 6 answers

What sort of functions do you allow? If you allow square roots then you can encode multiple inequalities in a single one:
A 0 B 0 ( A 2 A ) 2 + ( B 2 B ) 2 0

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