Stability in the Busemann-Petty and Shephard problems

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A comparison problem for volumes of convex bodies asks whether inequalities fₖ(ξ)≤ fₗ(ξ) for all ξ∈ Sⁿ⁻¹ imply that ₙ(K)≤ ₙ(L), where K,L are convex bodies in ⁿ, and fₖ is a certain geometric characteristic of K. By linear stability in comparison problems we mean that there exists a constant c such that for every >0, the inequalities fₖ(ξ)≤ fₗ(ξ)+ for all ξ∈ Sⁿ⁻¹ imply that (ₙ(K))n-1n≤ (ₙ(L))n-1n+c. We prove such results in the settings of the Busemann-Petty and Shephard problems and their generalizations. We consider the section function fₖ(ξ)=Sₖ(ξ)=ₙ₋₁(K∩ ξ) and the projection function fₖ(ξ)=Pₖ(ξ)=ₙ₋₁(K|ξ), where ξ⊥ is the central hyperplane perpendicular to ξ, and K|ξ is the orthogonal projection of K to ξ. In these two cases we prove linear stability under additional conditions that K is an intersection body or L is a projection body, respectively. Then we consider other functions fₖ, which allows to remove the additional conditions on the bodies in higher dimensions.