Intereting Posts

A ring with few invertible elements
Solve this integral:$\int_0^\infty\frac{\arctan x}{x(x^2+1)}\mathrm dx$
How prove $e^x|f(x)|\le 2$ if $f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$
Continuous functions and uncountable intersections with the x-axis
geometric multiplicity= algebraic multiplicity for a symmetric matrix
Pseudo-inverse of a matrix that is neither fat nor tall?
Good book for self study of a First Course in Real Analysis
Perfect square then it is odd
Pigeonhole principle and finite sequences
How to solve a complex polynomial?
If $a^2$ divides $b^2$, then $a$ divides $b$
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How many automorphisms of $\Bbb Z \oplus \Bbb Z_2$
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Some regularity in the prime decomposition

**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.

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**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.

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