| Summary: | Convex polygonal lines with vertices in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">Z</mi><mo>+</mo><mn>2</mn></msubsup></semantics></math></inline-formula> and endpoints at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>=</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mo>(</mo><msub><mi>n</mi><mn>1</mn></msub><mo>,</mo><msub><mi>n</mi><mn>2</mn></msub><mo>)</mo><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mn>2</mn></msub><mo>/</mo><msub><mi>n</mi><mn>1</mn></msub><mo>→</mo><mi>c</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, under the scaling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>n</mi><mn>1</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></semantics></math></inline-formula>, have limit shape <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>γ</mi><mo>*</mo></msup></semantics></math></inline-formula> with respect to the uniform distribution, identified as the parabola arc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msqrt><mrow><mi>c</mi><mrow><mspace width="0.222222em"></mspace><mspace width="-0.166667em"></mspace></mrow><mo>(</mo><mn>1</mn><mo>−</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo></mrow></msqrt><mo>+</mo><msqrt><msub><mi>x</mi><mn>2</mn></msub></msqrt><mo>=</mo><msqrt><mi>c</mi></msqrt></mrow></semantics></math></inline-formula>. This limit shape is universal in a large class of so-called multiplicative ensembles of random polygonal lines. The present paper concerns the inverse problem of the limit shape. In contrast to the aforementioned universality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>γ</mi><mo>*</mo></msup></semantics></math></inline-formula>, we demonstrate that, for any strictly convex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>C</mi><mn>3</mn></msup></semantics></math></inline-formula>-smooth arc <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>⊂</mo><msubsup><mi mathvariant="double-struck">R</mi><mo>+</mo><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> started at the origin and with the slope at each point not exceeding <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>90</mn><mo>∘</mo></msup></semantics></math></inline-formula>, there is a sequence of multiplicative probability measures <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>P</mi><mrow><mi>n</mi></mrow><mi>γ</mi></msubsup></semantics></math></inline-formula> on the corresponding spaces of convex polygonal lines, under which the curve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> is the limit shape.
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