TheoremBase

Let V be a real vector space, p: V o \mathbb{R} sublinear, U \subseteq V a linear subspace, and f: U o \mathbb{R} linear with f \le p on U. For any v_0 \in V \setminus U, define

\alpha := \sup{x \in U} \{\, f(x) - p(x+v_0) \,\},\qquad eta := \inf_{x \in U} \{\, p(x - v_0) - f(x) \,\}.

Then $\alpha \le eta$ . For any a \in [\alpha,eta], the formula

ilde f(x + t v_0) := f(x) + t a \quad (x \in U,\ t \in \mathbb{R})

extends f to the subspace U \oplus \mathbb{R} v_0 and satisfies $ilde f \le p$ on U \oplus \mathbb{R} v_0.

One-step extension under a sublinear bound

One-step extension under a sublinear bound

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