Rigid-flexible values for embeddings of four-dimensional ellipsoids into almost-cubes

Abstract We consider the embedding function c b ( a ) describing the problem of symplectically embedding an ellipsoid E (1, a ) into the smallest possible scaling by λ &gt; 1 of the polydisc P (1, b ). In particular, we calculate rigid-flexible values, i.e. the minimum a such that for E (1, a ′) with a ′ &gt; a , the embedding problem is determined only by the volume. For 1 &lt; b &lt; 2 with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>b</m:mi> <m:mo>≠</m:mo> <m:msup> <m:mfenced open="(" close=")"> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:mfrac> </m:mfenced> <m:mn>2</m:mn> </m:msup> </m:math> $b \neq\left(\frac{n+1}{n}\right)^2$ for all n ∈ ℕ, we find that these values vary piecewise smoothly. As Jin and Lee have analyzed the packing stability in the case b &gt; 2, our results complete the story outside of a discrete set.