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lambda-Convex Body

A lambda-convex body in Euclidean space R^n, for lambda>0, is a compact convex body with nonempty interior representable as an intersection of closed balls of radius 1/lambda. This is a curvature constraint stronger than ordinary convexity. A lambda-convex lens is the intersection of two such balls (Borisenko and Drach 2014, Drach and Tatarko 2023).

The Borisenko conjecture states that these lenses minimize volume at prescribed surface area. Drach and Tatarko (2026) proved the assertion in every dimension n>=2. For constant-curvature spaces, the analogous definition uses regions bounded by hypersurfaces of constant normal curvature lambda. In hyperbolic space these are not always ordinary geodesic balls.

See also

Borisenko Conjecture, Convex Body, Lens

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References

Borisenko, A. A. and Drach, K. D. "Isoperimetric Inequality for Curves with Curvature Bounded Below." Math. Notes 95, 590-598, 2014. https://doi.org/10.1134/S0001434614050034.Drach, K. and Tatarko, K. "Reverse Isoperimetric Problems under Curvature Constraints." 4 Mar 2023. https://arxiv.org/abs/2303.02294.Drach, K. and Tatarko, K. "A Complete Solution to the Reverse Isoperimetric Problem for lambda-Convex Bodies." 29 Oct 2026. https://arxiv.org/abs/2610.40291.

Cite this as:

Weisstein, Eric W. "lambda-Convex Body." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Lambda-ConvexBody.html

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