Geometry of spaces of real polynomials of degree at most \(n\) (Q1687693)

While the study of spaces of homogeneous polynomials over Banach spaces has been a very active area of research over the last decades, the non-homogeneous case has been scarcely looked into. The first systematic study of spaces of polynomials of degree at most \(n\) was addressed by \textit{C. Boyd} and \textit{A. Brown} in [J. Math. Anal. Appl. 429, No. 2, 1271--1290 (2015; Zbl 1325.46048)]. Now they continue this path by focusing on several geometric aspects of the space \(\mathcal P_I(^{\le n}E)\) of integral polynomials of degree at most \(n\). A predual of \(\mathcal P_I(^{\le n}E)\) can be represented by a sum of injective tensor products \((\bigoplus_{j=0}^n\bigotimes_{j,s} E, \|\cdot\|_\varepsilon)\) or, isometrically, by the space \(\mathcal P_A(^{\le n}E)\) of approximable polynomials of degree at most \(n\). In the article under review, several Šmul'yan-type theorems are presented, describing points of Gâteaux and Fréchet differentiability of the norm of \((\bigoplus_{j=0}^n\bigotimes_{j,s} E, \|\cdot\|_\varepsilon)\), resp. of \(\mathcal P_A(^{\le n}E)\), resp. of \(\mathcal P(^{\le n}E)\). For real Banach spaces \(E\), the authors describe the sets of extreme points (and weak\(^*\) exposed points) of the closed unit ball of \(\mathcal P_I(^{\le n}E)\). Most of the results of this interesting article can be seen as non-homogeneous versions of what was developed by \textit{C. Boyd} and \textit{R. A. Ryan} in [J. Funct. Anal. 179, No. 1, 18--42 (2001; Zbl 0977.46017)]. Anyway, there are surprising differences between the homogeneous and the non-homogeneous cases that the authors carefully point out.