We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N

STEM outreach evaluations often rely on full-sample averages and general enjoyment metrics, overlooking how specific pedagogical factors relate to students without a consolidated vocation. This study evaluates the “Desayuna Ingeniería” (Breakfast Engineering) STEM outreach programme to identify intervention variables associated with perceived increases in pre-university students’ vocational interest in industrial engineering. In the retrospective survey (N = 584), 48.8% of participants reported increased interest after the visit. Focusing specifically on the target subsample of students with low or moderate prior interest, who represent the “undecided” group (n = 408), binary logistic regression models examined how active workshop participation, overall workshop satisfaction, and mentor gender configuration were associated with perceived vocational change. Active workshop participation showed a positive predictive trend that did not reach the prespecified significance threshold (p = 0.056), overall workshop satisfaction showed no statistically significant association (p = 0.697), and evidence regarding mentor gender configuration was inconclusive. Exploratory gender-stratified analyses suggested that perceived increases were primarily associated with baseline interest in female students, and with baseline interest alongside a positive, although non-significant, trend for active participation (p = 0.052) in male students. Brief interventions may benefit from prioritising hands-on involvement, whereas fostering girls’ baseline STEM motivation at earlier school stages remains a theoretical hypothesis for future longitudinal studies.

Given two sets of points $A$ and $B$ in a normed plane, we prove that there are two linearly separable sets $A'$ and $B'$ such that $\mathrm{diam}(A')\leq \mathrm{diam}(A)$, $\mathrm{diam}(B')\leq \mathrm{diam}(B)$, and $A'\cup B'=A\cup B.$ This extends a result for the Euclidean distance to symmetric convex distance functions. As a consequence, some Euclidean $k$-clustering algorithms are adapted to normed planes, for instance, those that minimize the maximum, the sum, or the sum of squares of the $k$ cluster diameters. The 2-clustering problem when two different bounds are imposed to the diameters is also solved. The Hershberger-Suri's data structure for managing ball hulls can be useful in this context.

We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N.

Let $X$ be a real normed space with unit sphere S. We prove that $X$ is an inner product space if and only if there exists a real number $\rho=\sqrt{(1+\cos\frac{2k\pi}{2m+1})/2}$, $(k=1,2,\ldots , m ;\:m=1,2,\ldots)$, such that every chord of $S$ that supports $\rho S$ touches $\rho S$ at its middle point. If this condition holds, then every point $u\in S$ is a vertex of a regular polygon that is inscribed in $S$ and circumscribed about $\rho S$.