Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 7 1 2026 02 01 Geometry of variable-exponent Bochner-Lebesgue spaces‎: ‎Dentability‎, ‎Radon-Nikodym property‎, ‎and uniform convexity 1 12 238085 10.30504/jims.2026.567288.1310 EN M. Yaremenko National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine. 0000-0002-9209-6059 Journal Article 2025 12 19 This paper provides a comprehensive study of geometric properties of variable-exponent Lebesgue-Bochner spaces $L^{p(\cdot)}(E<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>X)$<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>. <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>We establish that $L^{p(\cdot)}(E‎, ‎X)$ has the Radon-Nikodym property (RNP) if and only if $X$ does‎, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>under the condition $p_m > 1$<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>. <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>We investigate dentability of specific subsets<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>showing that while the unit ball inherits dentability from $X$<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>the set of simple functions may remain nondentable even when the space possesses RNP<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>. <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>The perseverance of uniform convexity and smoothness are shown when both $X$ and $L^{p(\cdot)}(E)$ have these properties<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>. <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>We introduce quantitative dentability moduli and relate the dentability modulus of $L^{p(\cdot)}(E,X)$ to that of $X$<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>showing the relevance to the measure of $E$<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>and the oscillation of $p(\cdot)$<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>. <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>The paper reveals how variable exponents create hybrid geometries mixing $L^1$ and $L^p$ behaviors<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>with implications for PDE theory<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>optimization<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>, <span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>and geometric analysis<span lang="AR-SA" style="font-family: 'Arial',sans-serif; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-bidi;">‎</span>. https://jims.ims.ir/article_238085_74858c10672f1866351aa1776e771dfb.pdf