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  "body": "\n## Statement\n\nThe following statement is referred to as Whitehead's \"Peak Reduction Lemma\", see Proposition 4.20 in  and Proposition 1.2 in:\n\nLet $w\\in F_n$. Then the following hold:\n\n1. If $[w]$ is not automorphically minimal, then there exists a Whitehead automorphism $\\tau\\in \\operatorname{Aut}(F_n)$ such that $\\|\\tau(w)\\|_X\u003c \\|w\\|_X$.\n1. Suppose that $[w]$ is automorphically minimal, and that another conjugacy class $[w']$ is also automorphically minimal. Then $\\operatorname{Aut}(F_n)w=\\operatorname{Aut}(F_n)w'$ if and only if $\\|w\\|_X=\\|w'\\|_X$ and there exists a finite sequence of Whitehead moves $\\tau_1,\\dots, \\tau_k\\in\\operatorname{Aut}(F_n)$ such that $$\\tau_k\\cdots \\tau_1(w)=w'$$ and $$\\|\\tau_i\\cdots\\tau_1(w)\\|_X=\\|w\\|_X \\text{ for } i=1,\\dots, k.$$\n\nPart (1) of the Peak Reduction Lemma implies that a conjugacy class $[w]$ is Whitehead minimal if and only if it is automorphically minimal.\n",
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