Almost norming functionals, Part 1

Let E be a real Banach space with the dual E^*. By the Hahn-Banach theorem, for every unit vector e\in E there exists a functional e^*\in E^* of unit norm such that e^*(e)=1. One says that e^* is a norming functional for e. In general, one cannot choose e^* so that it depends continuously on e. For example, the 2-dimensional space with \ell_1 norm does not allow such a continuous selection.

Fix \delta\in (0,1) and call a linear functional e^* almost norming for e if |e|=|e^*|=1 and e^*(e)\ge \delta. In any Banach space there exists a continuous selection of almost norming functionals.

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Expanding distances on a sphere

This came up in a discussion at AIM.

Let \mathbb S^{n-1}\subset \mathbb R^n be the unit sphere. Suppose that f\colon \mathbb S^{n-1}\to\mathbb R^n is a continuous map which does not decrease distances: that is, |f(x)-f(y)|\ge |x-y| for all x,y\in \mathbb S^{n-1}. By the generalized Jordan theorem the complement of f(\mathbb S^{n-1}) has two components, precisely one of which, denoted \Omega, is bounded.

Prove that \Omega contains an open ball of radius 1.

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Generalized Birthday Problem

Facebook currently allows up to 5000 friends. Assume that birthdays are uniformly distributed among 365 days. It is well-known that for a user with 23 friends there is a 50% chance of having to write ”Happy Birthday” on at least two walls in the same day.

How many facebook friends should one acquire to have a 50% chance of at least 10 birthdays falling on the same day? With 3286 friends this is guaranteed to happen, but one should expect that 50% probability will be reached with a substantially smaller number.

[Update] In 1989 Diaconis and Mosteller found an approximate solution that gives an approximate number of people required to have at least 50% probability of of at least k birthdays on the same day. The formula can be found on MathWorld and its output is the OEIS sequence A050255. For comparison, Diaconis and Mosteller quote exact values found by Bruce Levin. The exact solution is the sequence A014088, which begins thus:

1, 23, 88, 187, 313, 460, 623, 798, 985, 1181, 1385, 1596, 1813, 2035, 2263

So, with 1181 Facebook friends there is a 50% chance of having to write “Happy Birthday” at least ten times in the same day.