Volumetric bounds for intersections of congruent balls in Euclidean spaces

Reverse isoperimetric problems under curvature constraints

“…As we will see below, in R n , the lens 'wins' this competition asymptotically as n → ∞. Spindles also appear as solutions to other optimization problems with curvature constraints (see, e.g., [Be1,BD1]).…”

“…Much later, this result was generalized in [Dr4] to arbitrary λ-convex domains in 2-dimensional Alexandrov spaces of curvature bounded below, thus closing this question in dimension 2. Recently, Bezdek [Be1] showed that among all λconvex bodies in R n , n 2, the λ-convex lens has the smallest inradius for a given volume. Bezdek conjectured that the same result must be true if one considers λ-convex bodies of a given surface area [Be1,Conjecture 5].…”

“…Recently, Bezdek [Be1] showed that among all λconvex bodies in R n , n 2, the λ-convex lens has the smallest inradius for a given volume. Bezdek conjectured that the same result must be true if one considers λ-convex bodies of a given surface area [Be1,Conjecture 5]. We confirm this conjecture by establishing the following sharp and much more general result:…”