Let $X$ be a Banach space and $d$ be the induced metric. Let $S(x;r)$ denote the closed ball with radius $r$ at $x\in X$, that is,$S(x;r)=\lbrace y\in X\colon d(x,y)\le r\rbrace.$
Let $x,y\in X$ and define the sequence $\{f_n^x(y)\colon n\in\mathbf{N}\}$ where $f_n^x(y)=\Big[1-n\cdot\mathrm{dist}(y,S(x;r))\Big]^+.$ The plus sign denotes the positive part (see http://en.wikipedia.org/wiki/Positive_and_negative_parts) and $\mathrm{dist}(y,S(x;r))=\inf\{d(y,z)\colon z\in S(x;r)\}$
Does the sequence $\{f_n^x(y)\colon\in\mathbf{N}\}$ converge for $n\to\infty$ and what is the limit? My best guess is that this sequence converges to the characteristic function on the closed ball, but I don't know how to prove this.