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A closed convex set $Q$ in a local convex topological Hausdorff spaces $X$ is called locally nonconical (LNC) if for every $x,y\in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q)+\frac{1}{2}(y-x)\subset Q$. A set $Q$ is local cylindric (LC) if for $x,y\in Q$, $x\ne y$, $z\in (x,y)$ there exists an open neighbourhood $U$ of $z$ such that $U\cap Q$ (equivalently: $\mathrm{b}d\left(Q\right)\cap U$) is a union of open segments parallel to $[x,y]$. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication $\mathrm{L}NC\Rightarrow \mathrm{L}C$ was proved in general, while...

The aim of this paper is to investigate the local nonconicality of unit ball in Orlicz spaces, endowed with the Luxemburg norm. A closed convex set $Q$ in a locally convex topological Hausdorff space $X$ is called locally nonconical $\left(LNC\right)$, if for every $x,y\in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q)+(y-x)/2\subset Q$. The following theorem is established: An Orlicz space $L^{\varphi }\left(\mu \right)$ has an $LNC$ unit ball if and only if either $L^{\varphi }\left(\mu \right)$ is finite dimensional or the measure $\mu$ is atomic with a positive greatest lower bound and $\varphi$ satisfies...

The aim of this paper is to investigate the stability of the positive part of the unit ball in Orlicz spaces, endowed with the Luxemburg norm. The convex set $Q$ in a topological vector space is stable if the midpoint map $\Phi :Q\times Q\to Q$, $\Phi (x,y)=(x+y)/2$ is open with respect to the inherited topology in $Q$. The main theorem is established: In the Orlicz space $L^{\varphi }\left(\mu \right)$ the stability of the positive part of the unit ball is equivalent to the stability of the unit ball.