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

Question

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

aqueritztv · Accepted Answer

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  =  0If 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 · Accepted Answer

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

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

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