Let H V Rn, with H convex and V open, and suppose that : V Rn is C1. a) Show that if

Let H Š‚ V Š‚ Rn, with H convex and V open, and suppose that ɸ: V †’ Rn is C1.
a) Show that if E is a closed subset of H° and

then ˆˆh(x)/||h|| †’ 0 uniformly on E, as h †’ 0.
b) Show that if R is a closed rectangle in H° and S := (Dɸ(x))-l exists for some x ˆˆ R, then given ε > 0 there are constants δ > 0 and M > 0 and a function T(x, y) such that
S o ɸ(x) - 5 o ɸ(y) = x - y + T(x, y)
for x, y ˆˆ R, and ||T(x, y)|| c) Use parts a) and b) to prove that if Δɸ is nonzero on V, x ˆˆ H°, and ε is sufficiently small, then there exist numbers Cε > 0, which depend only on H, ɸ, n, and ε, and a δ > 0 such that Cε †’ 1 as ε †’ 0 and Vol(S o ɸ(Q)) d) Use part c) and Exercise 12.4.9 to prove that if Δɸ is nonzero on V and x ˆˆ H0, then given any sequence of cubes Qj which satisfy x ˆˆ Qj and Vol(Qj) †’ 0 as j †’ ˆž, it is also the case that Vol(ɸ (Qj))/|Qj| Δɸ (x)| as j †’ ˆž.

6h(x) := (x h) _ (x)-1(x)(h), for x E V and h small.