Example 21.4.556EB803309C2

Example 21.4.5.56EB803309C2 The hypotheses $K_ F=P$ and $K_ G=P'$ in the last assertion of proposition 21.4.2 cannot be replaced by $F\in \mathcal E(N_{\mathbb R},P)$ and $G\in \mathcal E(N_{\mathbb R},P')$: when $P$ has empty interior, membership in $\mathcal E(N_{\mathbb R},P)$ does not force a function to use every slope in $P$. Let $N_{\mathbb R}=M_{\mathbb R}=\mathbb R^2$ with the standard pairing, $P=[0,1]\times \{ 0\} $ and $P'=\{ 0\} \times [0,1]$. Both bodies have empty interior, and the zero function belongs to $\mathcal P(N_{\mathbb R},P)=\mathcal E(N_{\mathbb R},P)$ and to $\mathcal E(N_{\mathbb R},P')$, since $0\leq h_ P$ and $0\leq h_{P'}$. The sum of the zero function with itself is zero, whose slope body is $\{ 0\} $, whereas $P+P'=[0,1]^2$ has nonempty interior. Hence the zero function does not belong to $\mathcal E(N_{\mathbb R},P+P')$.