In K.Urai's paper "Fixed point theorems and the existence of economic equilibria based on conditions for local directions of mappings", he claimed the following in Lemma 17:
Statement: Let $X\subset \mathbb{R}^n$ be a nonempty compact convex subset, and $f:X\to X$ be any function. Then $0$ is in the convex hull of the set $\{f(x)-x|x\in X\}$.
However, his proof is false. He used the following wrong argument: if $C\subset \mathbb{R}^n$ is a nonempty convex subset and $0\notin C$, then there is $p\in \mathbb{R}^n$ such that the inner product $(p,c)>0$ for any $c\in C$.
But the Statement may be true, through. For example, I think I can prove it when $X$ is the closed unit ball or a simplex of $\mathbb{R}^n$. Is the Statement really true for general $X$?
Here is a proof via induction over the dimension of (the affine hull of) $X$: Suppose that $0$ does not belong to the convex hull of $\{f(x) - x \mid x \in X\}$. Then, there exists $p \in \mathbb R^n \setminus \{0\}$ with $$ p^\top f(x) \ge p^\top x \qquad\forall x \in X. $$ We set $$ \hat X = \{ x \in X \mid p^\top x = \max_{y \in X} p^\top y\}.$$ Clearly, $\hat X$ is nonempty, closed and convex and the dimension is less than the dimension of $X$. Moreover, the above inequality ensures $$ p^\top f(x) \ge p^\top x = \max_{y \in X} p^\top x \qquad\forall x \in \hat X,$$ i.e., $f \colon \hat X \to \hat X$. By the induction hypothesis, $0$ belongs to the convex hull of $$\{f(x) - x \mid x \in \hat X\}$$ and this is a subset of the convex hull of $$\{f(x) - x \mid x \in X\}.$$ This finishes the induction step.