The (Un)equal Tangents Problem

Given a point A outside of a closed strictly convex plane curve γ, there are two tangent segments from A to γ, the left and the right ones, looking from point A. Problem: Does there exist a curve γ such that one can walk around it so that, at all moments, the right tangent segment is smaller than the left one? In other words, does there exist a pair of simple closed curves, γ and Γ, the former strictly convex, the latter containing the former in its interior, such that for every point A of Γ the right tangent segment to γ is smaller than the left one? Over the years, I have polled numerous colleagues, mostly as a dinner table topic. Most of them thought that the answer was negative, and quite a few tried to provide a proof, but each attempt had a flaw. I invite the reader to think about this question too before reading any further. Up until recently, I have believed that for any oval1 γ and every closed curve Γ going around γ, there existed a point A ∈ Γ from which the tangent segments to γ were equal. In fact, I conjectured in [7] that there existed at least four such points. Figure 1 illustrates the situation for an ellipse. The extensions of the axes partition the plane into four quadrants marked + and − according to the sign of the difference between the left and the right tangent segments. The axes themselves are the locus of points from which the tangent segments