Convex Decomposition

[selman.ipek]

Hi All, I have the vertices for two polytopes \( P_1 \) and \( P_2 \). Mathematically I know that that \( P_2 \hookrightarrow P_1 \) and would like to decompose the vertices of \( P_2 \) in terms of embedding \( P_1 \). Any suggestions on how to do this? Can polymake help? I couldn't find anything in the manual. Alternative software suggestions also welcome, thanks!

[joswig]

Sorry, I don't understand the question.

Is it that P_2 is a subpolytope of P_1, and you want to find those vcertices of P_1 which are not vertices of P_2?

[selman.ipek]

Hi, thanks for the response. The problem is that since P2 is a properly embedded in P1 I would like to express the vertices of P2 as a convex sum of the vertices of P1.

[joswig]

That does not have unique solutions. Instead the set of such solutions forms a polyhedron in the parameter space (= coefficients of the linear combinations).

[selman.ipek]

Thank you again for the response. For my purposes any decomposition might suffice, but this suggestion about the coefficients may be quite useful

[opfer]

Isn't this just solving a linear program \( V_1 \cdot x = v_{2,i} \), \( x>=0 \), \( \sum_j x_j = 1 \) for each vertex \( v_{2,i} \) of \( P_2 \) (\( V_1 \) being the Matrix containing the vertices of \( P_1 \), \( x \) being the coefficients of a possible convex combination)? Any LP solver (including Polymake) should be able to do this.

Best regards,
Thomas