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In this paper, for d ≥ 1 and s ∈ (0 , d 2) , we study the Bianchi–Egnell quotient Q (f) = inf f ∈ H ˙ s (R d) \ B ‖ (- Δ) s / 2 f ‖ L 2 (R d) 2 - S d , s ‖ f ‖ L 2 d d - 2 s (R d) 2 dist H ˙ s (R d) (f , B) 2 , f ∈ H ˙ s (R d) \ B , where S d , s is the best Sobolev constant and B is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when d = 1 , there is a neighborhood of B on which the quotient Q (f) is larger than the lowest value attainable by sequences converging to B . This behavior is surprising because it is contrary to the situation in dimension d ≥ 2 described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for d = 1 , Q (f) has no minimizer on H ˙ s (R d) \ B , which again would be contrary to the situation in d ≥ 2 . As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every d ≥ 1 . For d ≥ 2 , this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For d = 1 we make some numerical observations which support the conjecture stated above. [ABSTRACT FROM AUTHOR]

Among probability measures on d-dimensional real projective space, one which maximizes the expected angle arccos (x | x | · y | y |) between independently drawn projective points x and y was conjectured to equidistribute its mass over the standard Euclidean basis { e 0 , e 1 , ... , e d } by Fejes Tóth (Acta Math Acad Sci Hung 10:13–19, 1959. https://doi.org/10.1007/BF02063286). If true, this conjecture evidently implies the same measure maximizes the expectation of arccos α (x | x | · y | y |) for any exponent α > 1 . The kernel arccos α (x | x | · y | y |) represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this milder conjecture in a non-empty range α > α Δ d ≥ 1 , and establish uniqueness of the resulting maximizer μ ^ up to rotation. We show μ ^ no longer maximizes when α < α Δ d . At the endpoint α = α Δ d of this range, we show another maximizer μ must also exist which is not a rotation of μ ^ . For the continuous version of the conjecture, an "Appendix A" provided by Bilyk et al in response to an earlier draft of this work combines with the present improvements to yield α Δ d < 2 . The original conjecture α Δ d = 1 remains open (unless d = 1 ). However, in the maximum possible range α > 1 , we show μ ^ and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the L ∞ -Kantorovich–Rubinstein–Wasserstein metric d ∞ from optimal transportation; the same is true for any measure μ which is mutually absolutely continuous with respect to μ ^ , but the size of the ball depends on α , d , and ‖ d μ ^ d μ ‖ ∞ . [ABSTRACT FROM AUTHOR]