A Convex Function is Bounded Above near a Point by its Values at Coordinate Neighbours

Let $n$ be a natural number with $1\le n$, let $[n]$ be the initial segment determined by $n$, and let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, with the absolute value written $|\cdot|$. Let $C$ be a convex subset of Euclidean space $\mathbb{R}^{n}$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and let $u:C\to\mathbb{R}$ be convex on $C$.

For $i\in[n]$ let $e^{(i)}\in\mathbb{R}^{n}$ be the point whose $i$th coordinate is $1$ and whose $j$th coordinate is $0$ for every $j\in[n]$ with $j\ne i$. Let $x_{0}\in\mathbb{R}^{n}$, let $r\in\mathbb{R}$ with $0<r$, and suppose that

$$x_{0}\in C,\qquad x_{0}+r\,e^{(i)}\in C,\qquad x_{0}-r\,e^{(i)}\in C\qquad\text{for every }i\in[n].$$

Let $M\in\mathbb{R}$ satisfy

$$u(x_{0})\le M,\qquad u\bigl(x_{0}+r\,e^{(i)}\bigr)\le M,\qquad u\bigl(x_{0}-r\,e^{(i)}\bigr)\le M\qquad\text{for every }i\in[n].$$

Then every $x\in\mathbb{R}^{n}$ with

$$\sum_{i=1}^{n}\bigl|x_{i}-(x_{0})_{i}\bigr|\le r,$$

the sum being the finite sum of $\mathbb{R}$, lies in $C$ and satisfies $u(x)\le M$.