In this talk, new results for using the $1$-dimensional extension property to determine analyticity are discussed. The case under consideration is what has been referred to as the strip problem. Let $C$ be a strictly convex closed curve in the complex plane whose horizontal translates $C_t$ fill out a strip $S.$ Given a function $f(z)$ defined on the strip, such that the restriction of $f$ to each $C_t$ has a holomorphic extension to the domain $D_t,$ show that $f$ is holomorphic. Very general theorems of this type, including results in several variables, have been proved for real analytic functions; some more specialized theorems have been proved for the strip problem, for functions continuous on the closed strip. The current results are for curves which are very general, but not completely generic, perturbations of an ellipse. The motivation for this research was to demonstrate that theorems of this type can be proved for $L^p$ functions. Also, restricting to the case of the strip allows one to consider rather difficult geometrical problems; success here may lead to new theorems in several variables. The techniques involve a very detailed examination of a certain complexified problem, as well as some specialized wedge extension theorems.