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In this paper, we explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of  F m  ×ℤ  n   onTree   ×   ℝ  n${\mathrm {Tree}\times \mathbb {R}^n}$   , every quasiconvex subgroup of  F m  ×ℤ  n   is convex. Further, we show that the Cartan–Hadamard theorem can be used to show that locally convex subsets of complete and connected CAT(0) spaces are convex. Finally, we show that the quasiconvex subgroups of  F m  ×ℤ  n   are precisely those of the form  A × B , whereA  ≤   F  m${A\le F_m}$    is finitely generated, andB  ≤   ℤ  n${B\le \mathbb {Z}^n}$   .

On convex hulls and the quasiconvex subgroups of F m ×ℤ n