Let D be a convex domain with smooth boundary in complex space and let f be a continuous function on the boundary of D. Suppose that f holomorphically extends to the extremal discs tangent to a convex subdomain of D. We prove that f holomorphically extends to D. The result partially answers a conjecture by Globevnik and Stout of 1991.
"Extremal discs and holomorphic extension from convex hypersurfaces." Ark. Mat. 45 (1) 1 - 13, April 2007. https://doi.org/10.1007/s11512-006-0016-7