Quasiconvex and lower semicontinuous function

Definition 1. Let $X$ be a Banach space and $f:X\rightarrow\overline{\mathbb{R}}$. The function $f$ is said to be proper if $f(x)>-\infty$ for all $x\in X$ and there exists $x_0\in X$ such that $f(x_0)<\infty$.

Definition 2. Let $X$ be a Banach space and $f:X\rightarrow\overline{\mathbb{R}}$. The function $f$ is said to be quasiconvex if for every $\alpha\in\mathbb{R}$ the level set
$$
L_\alpha:=\{x\in X: f(x)\leq\alpha\}
$$
is convex.

Definition 3. Let $X$ be a Banach space and $f:X\rightarrow\overline{\mathbb{R}}$. The function $f$ is said to be lower semicontinuous if for every $\alpha\in\mathbb{R}$ the level set
$$
L_\alpha:=\{x\in X: f(x)\leq\alpha\}
$$
is closed.

Question. Let $X$ be a Banach space and $\varphi:X\rightarrow\overline{\mathbb{R}}$ a proper, lower semicontinuous, and quasiconvex function. Let $u,v\in X$ such that $u\ne v$ and $\varphi(u)\leq\varphi(v)$. Suppose that there exists $v^*\in X^*$ such that
$$
\sup_{z\in ]u,v[}\{\varphi(z)+\langle v^*,z\rangle\}>\max\{\varphi(u)+\langle v^*,u\rangle,\varphi(v)+\langle v^*,v\rangle\}.
$$
I would like to know whether there exists $w\in]u,v[$ such that
$$
\forall r>0,\exists w_r\in B_r(w)\bigcap]w,v[: \varphi(w)+\langle v^*,w\rangle\geq \varphi(w_r)+\langle v^*,w_r\rangle.
$$
Thank you for all comments and helping.

Solutions Collecting From Web of "Quasiconvex and lower semicontinuous function"