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.

We introduce some practical calculation of the Riesz angles in Orlicz sequence spaces equipped with Luxemburg norm and Orlicz norm. For an $N$-function $\Phi \left(u\right)$ whose index function is monotonous, the exact value $a\left(l^{\left(\Phi \right)}\right)$ of the Orlicz sequence space with Luxemburg norm is $a\left(l^{\left(\Phi \right)}\right)=2^{\frac{1}{C_{\Phi }^0}}$ or $a\left(l^{\left(\Phi \right)}\right)=\frac{{\Phi }^{-1}\left(1\right)}{{\Phi }^{-1}\left(\frac{1}{2}\right)}$. The Riesz angles of Orlicz space $l^{\Phi }$ with Orlicz norm has the estimation $max(2{\beta }_{\Psi }^0,2{\beta }_{\Psi }^{\text{'}})\le a\left(l^{\Phi }\right)\le \frac{2}{{\theta }_{\Phi }^0}$.

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...