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S(X)/P(X)<2 S(Y)/P(Y), convex polygons (2019 Novosibirsk Oral Geo Oly 9.7)

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April 28, 2021
geometrygeometric inequalityconvex polygon

Problem Statement

Denote X,YX,Y two convex polygons, such that XX is contained inside YY. Denote S(X)S (X), P(X)P (X), S(Y)S (Y), P(Y)P (Y) the area and perimeter of the first and second polygons, respectively. Prove that S(X)P(X)<2S(Y)P(Y). \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.
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