Flattening the Curve

We quantify flattening the curve under the assumption of a soft quarantine in the spread of a contagious viral disease in a society. In particular, the maximum daily infection rate is expected to drop by twice the percentage drop in the virus reproduction number. The same percentage drop is expected for the maximum daily hospitalization or fatality rate. A formula for the expected maximum daily fatality rate is given

Uniform Eberlein Compacta and Uniformly Gâteaux Smooth Norms

* Supported by grants: AV ĈR 101-95-02, GAĈR 201-94-0069 (Czech Republic) and NSERC 7926 (Canada).It is shown that the dual unit ball BX∗ of a Banach space X∗ in its weak star topology is a uniform Eberlein compact if and only if X admits a uniformly Gâteaux smooth norm and X is a subspace of a weakly compactly generated space. The bidual unit ball BX∗∗ of a Banach space X∗∗ in its weak star topology is a uniform Eberlein compact if and only if X admits a weakly uniformly rotund norm. In this case X admits a locally uniformly rotund and Fréchet differentiable norm. An Eberlein compact K is a uniform Eberlein compact if and only if C(K) admits a uniformly Gˆateaux differentiable norm

Strong subdifferentiability of norms and geometry of Banach spaces

summary:The strong subdifferentiability of norms (i.e\. one-sided differentiability uniform in directions) is studied in connection with some structural properties of Banach spaces. It is shown that every separable Banach space with nonseparable dual admits a norm that is nowhere strongly subdifferentiable except at the origin. On the other hand, every Banach space with a strongly subdifferentiable norm is Asplund

A note on lattice renormings

summary:It is shown that every strongly lattice norm on $c_0(\Gamma)$ can be approximated by $C^\infty$ smooth norms. We also show that there is no lattice and G\^ateaux differentiable norm on $C_0[0,\omega_1]$

Weakly Compact Generating and Shrinking Markusevic Bases

2000 Mathematics Subject Classification: 46B30, 46B03.It is shown that most of the well known classes of nonseparable Banach spaces related to the weakly compact generating can be characterized by elementary properties of the closure of the coefficient space of Markusevic bases for such spaces. In some cases, such property is then shared by all Markusevic bases in the space

A note on extreme points of C∞C^\infty-smooth balls in polyhedral spaces

[EN] Morris (1983) proved that every separable Banach space $X$ that contains an isomorphic copy of $c_0$ has an equivalent strictly convex norm such that all points of its unit sphere $S_X$ are unpreserved extreme, i.e., they are no longer extreme points of $B_{X^{**}}$. We use a result of Hájek (1995) to prove that any separable infinite-dimensional polyhedral Banach space has an equivalent $C^{\infty}$-smooth and strictly convex norm with the same property as in Morris' result. We additionally show that no point on the sphere of a $C^2$-smooth equivalent norm on a polyhedral infinite-dimensional space can be strongly extreme, i.e., there is no point $x$ on the sphere for which a sequence $(h_n)$ in $X$ with $\Vert h_n\Vert\not \to 0$ exists such that $\Vert x\pm h_n\Vert\to 1$.The first author’s research was supported by Ministerio de Econom´ıa y Competitividad and FEDER under project MTM2011-25377 and the Universitat Polit`ecnica de Val`encia. The second author’s research was supported by Ministerio de Econom´ıa y Competitividad and FEDER under project MTM2011-22417 and the Universitat Polit`ecnica de Val`enciaGuirao Sánchez, AJ.; Montesinos Santalucia, V.; Zizler, V. (2015). A note on extreme points of $C^\infty$-smooth balls in polyhedral spaces. Proceedings of the American Mathematical Society. 143(8):3413-3420. https://doi.org/10.1090/S0002-9939-2015-12617-2S34133420143