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Let $(V, \lVert \cdot \rVert)$ be a normed space. For any $A \subseteq V$, let $O(A)$ be the intersection of all closed balls containing $A$, or more precisely, let $O \colon 2^V \to 2^V$ be defined b …

Let $(X, \lVert \cdot \rVert)$ be a separable normed space. Can we always guarantee that there is a nonempty compact set $K \subseteq B_X$, where $B_X$ is a closed unit ball in $X$ such that: $$\foral …

A (real) normed space $(V, \lVert \cdot \rVert_V)$ is called strictly convex if for all $x, y \in V \setminus \{ 0 \}$ we have \begin{equation} \lVert x + y \rVert_V = \lVert x \rVert_V + \lVert y \rV …

For a normed space $(V, \lVert\cdot\rVert_V)$ let us define: \begin{equation} \forall x, y \in V \quad [0,1] \mapsto \gamma_x^y (t) = (1-t)x + ty. \end{equation} I would like to ask whether the follo …

Let $(V, \lVert \cdot \rVert)$ be a normed space. Let us consider the set $C = [-1,1]^{\dim V}$. The boundary of this set consists of closed subsets $B_i$ (indexed by some set $I$) of affine hyperplan …

Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} …

Let $(X, \lVert \cdot \rVert_X)$, $(Y, \lVert \cdot \rVert_Y)$ be two Banach spaces. A function $P \colon X \to Y$ such that there exists $n \in \mathbb{N}$ such that for all $i \in \{ 0, \ldots, n \} …

Let $\Omega \subseteq \mathbb{R}^M$ be a measurable subset of positive measure. R. A. Adams and J. Fournier in their article have proven that if $\Omega$ satisfies the so-called weak cone property, t …

Since in a finite-dimensional normed space all norms are equivalent, the terms $\nabla f$, $\nabla g$, and $\nabla^2 g$ wouldn't change regardless of the used norm. …