Let $C$ be a convex subset of a vector space $V$. We consider two definitions of the intrinsic core of $C$.
Definition 1. The intrinsic core of $C$ consists all points $c\in C$ such that for every $c^\prime\in\text{aff}(C)\setminus\{c\}$ we have $(c,c^\prime)\bigcap C\ne \emptyset$.
Definition 2. The intrinsic core of $C$ consists all points $c\in C$ such that for every $c^\prime\in C\setminus\{c\}$ there exists $c^{\prime\prime}\in C$ such that $c\in (c^\prime, c^{\prime\prime})$.
Here, $\text{aff}(C)$ is the affine hull of C and $(a,b):=\{(1-t)a+tb:t\in(0,1)\}$.
Observe that Definition 1 implies Definition 2. Indeed suppose that $c\in C$ is a point in the intrinsic core of $C$ in the Definition 1. Let $c^\prime\in C\setminus\{c\}$. Let $t\in (0,1)$ and $c_t=c^\prime+(1/t)(c-c^\prime)$. Then $c\in (c^\prime, c_t)$ and $c_t\in\text{aff}(C)\setminus\{c\}$. From Definition 1 there exist $c^{\prime\prime}\in (c,c_t)\bigcap C$. Since $c^{\prime\prime}\in (c,c_t)$ and $c\in (c^\prime, c_t)$, we have $c\in(c^\prime,c^{\prime\prime})$. Hence, $c$ is the point in the intrinsic core of $C$ in the Definition 2.
Could we prove that Definition 2 implies Definition 1?
Thank you for all kind help.
Let $c$ be in the intrinsic core of $C$ according to definition 2. Let $c' \in aff(C)$ be given. Then, $$c' = c + \sum_{i=1}^n \lambda_i \, (c_i - c)$$ for $c_i \in C$. According to definition 2, there is $T > 0$ such that $c + t \, (c_i-c) \in C$ for all $|t| \le T$. W.l.o.g. we can assume $\lambda_i \ge 0$ (otherwise, replace $c_i$ by $\hat c_i = c + T\,(c-c_i) \in C$ and $\lambda_i$ accordingly). By taking convex combinations,
$$c + t \, (c' - c) = c + \sum_{i=1}^n \lambda_i \, t \, (c_i-c) = \sum_{i=1}^n \lambda_i \, t \, c_i \in C$$ for $t = 1/\sum_{i=1}^n\lambda_i$.
Hence, $c$ is in the intrinsic core of $C$ according to definition 1.