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We are concerned with the Calder\'on inverse inclusion problem, where oneintends to recover the shape of an inhomogeneous conductive inclusion embeddedin a homogeneous conductivity by the associated boundary measurements. Weconsider the highly challenging case with a single partial boundarymeasurement, which constitutes a long-standing open problem in the literature.It is shown in several existing works that corner singularities can help toresolve the uniqueness and stability issues for this inverse problem. In thispaper, we show that the corner singularity can be relaxed to be a certainhigh-curvature condition and derive a novel local unique determination result.To our best knowledge, this is the first (local) uniqueness result indetermining a conductive inclusions with general smooth shapes by a single(partial) boundary measurement.

On Calder\'on's inverse inclusion problem with smooth shapes by a single partial boundary measurement