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This paper shows that Gr\"obner walks aiming for the elimination of variablesfrom a polynomial ideal can be terminated much earlier than previously known.To this end we provide an improved stopping criterion for a known Gr\"obnerwalk algorithm for the elemination of variables. This results from two newgeometric insights on Gr\"obner fans: We show that for any given ideal I\subset K[x_1, ..., x_n] the collection of Gr\"obner cones corresponding toI-specific elimination orders may contain Gr\"obner cones in the relativeinterior of the positive orthant. Moreover we prove that the correspondingGr\"obner cones form a star-shaped region (the center being the set of alluniversal elimination vectors) which contrary to first intuition in general isnot convex.

Ideal-specific elimination orders form a star-shaped region