proving that $\text{aff}C-\text{aff}C\subset\text{aff}\,(C-C)$

Question

In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that $\text{rge}\,A\subset\text{aff}\,C$ and for $\epsilon>0$ claims that $\epsilon^{-1}(C-\text{rge}\,A)\subset\text{aff}\,(C-C)$, that I can't verify it.

It's clear that $\epsilon^{-1}(C-\text{rge}\,A)\subset\epsilon^{-1}(\text{aff}\,(C)-\text{aff}\,(C))$, so if we let $\omega\in\epsilon^{-1}(\text{aff}\,C-\text{aff}\,C)$, then $$\omega=\sum_{i=1}^m\epsilon^{-1}\lambda_iu_i-\sum_{i=1}^m\epsilon^{-1}\mu_iv_i=\sum_{i=1}^m\lambda_i(\epsilon^{-1}u_i-\epsilon^{-1}\frac{\mu_i}{\lambda_i}v_i),u_i,v_i\in C,\sum_{i=1}^m\lambda_i=\sum_{i=1}^m\mu_i=1$$ which is in $\text{aff}(C-C)$ iff $\epsilon^{-1}u_i,\epsilon^{-1}\frac{\mu_i}{\lambda_i}v_i\in C$, but we just have that $C$ is a convex set.

Answer 1

Here is one proof of $(\operatorname{aff} C - \operatorname{aff} C) \subset \operatorname{aff} (C - C)$.

Note that $S$ is affine iff $S$ can be written as $\{x_0\}+L$ for some linear space $L$.

Let $\operatorname{aff} C = \{x_0\} +L$. Then $\operatorname{aff} C - \operatorname{aff} C = \{x_0\} +L + \{-x_0\} +(-L) = L$, hence $\operatorname{aff} C - \operatorname{aff} C$ is the corresponding linear space.

Now let $c_0 \in C$ and note that $\operatorname{aff} (C - \{c_0\}) \subset \operatorname{aff} (C - C)$ and so $\operatorname{aff} (C) - \{c_0\} \subset \operatorname{aff} (C - C)$. Hence $\{x_0-c_0\} +L = L \subset \operatorname{aff} (C - C)$, from which the result follows.

Answer 2

We prove that if $A$ and $B$ are linear manifolds then $A+B$ is also a linear manifold. Indeed, let $x,y\in A+B$ and $\lambda\in\mathbb{R}$. Then there exist $x_a,y_a\in A$ and $x_b,y_b\in B$ such that $x_a+x_b=x$ and $y_a+y_b=y$. Then $$ \lambda x+(1-\lambda)y=\lambda(x_a+x_b)+(1-\lambda)(y_a+y_b)=[\lambda x_a+(1-\lambda)y_a]+[\lambda x_b+(1-\lambda)y_b]. $$ Since $A, B$ are linear manifolds, $\lambda x_a+(1-\lambda)y_a\in A$ and $\lambda x_b+(1-\lambda)y_b\in B$, and so $\lambda x+(1-\lambda)y\in A+B$. Therefore, $A+B$ is also a linear manifold.

We have $C\subset \text{aff}(C)$. Then $C-C\subset\text{aff}(C)-\text{aff}(C)$. Hence $\text{aff}(C)-\text{aff}(C)$ is a linear manifold containing $C-C$. It follows that $\text{aff}(C-C)\subset\text{aff}(C)-\text{aff}(C)$ ($\text{aff}(C-C)$ is the smallest linear manifold containing $C-C$).