Let \(J\) be a smooth Jordan curve in \(\mathbb{R}^2\) satisfying the twin ball condition: there is \(\alpha>0\) such that for every \(z\in J\) there are two points \(a_z,b_z\) lying on different sides of \(J\) and such that \(\overline{B} (z_2,\alpha)\cap J=\{z\}= \overline{B} (b_z,\alpha)\cap J\), where \(\overline{B} (q,\alpha)\) denotes the closed ball in \(\mathbb{R}^2\) of center \(q\in \mathbb{R}^2\) and radius \(\alpha>0\). For \(u\in \mathbb{R}^2\) let \(\rho(u,J)= \inf\{| u-z|: z\in J\}\). The authors show that under the above hypotheses there is \(r>0\) such that if \(u\in \mathbb{R}^2\) satisfies \(\rho(u,J)< r\) then there is \(v\in J\) with \(| u-v|<| u-z|\) for all \(z\in J\setminus \{v\}\). The proof of this result is based on methods of constructive analysis, without using the fact that a continuous real valued function attains its minimum on a compact set.