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A new characterization of convexity of a planar domain is obtained. Its derivation involves two classical facts: the Varadhan's formula, expressing the distance function with respect to the domain's boundary via real-valued solutions of the modified Helmholtz equation, and the convexity of a planar domain in which the distance function is superharmonic
Н. В. Кузнецов (Thu,) studied this question.
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