Clarification for the greedy algorithm validity

[kartynnik]

Hello! Thank you for these fascinating research results.
I guess this is a silly question, but still: Could you please provide an argument for the greediness in the algorithm used in the proximal operator computation? Precisely, what I am interested in is why, once reaching an edge of the polyhedron, we should move along that edge (not considering the other face) and why the edge direction is given by averaging the "clashed" components of the previous direction? The rest of the algorithm is well-grounded and clear to me.
Thanks in advance!