| Summary: | Recently, Ledoux, Nair, and Wang proved that the Fisher information along the heat flow is log-convex in dimension one, that is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></mstyle><mo form="prefix">log</mo><mrow><mo stretchy="false">(</mo><mi>I</mi><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>t</mi></msub></semantics></math></inline-formula> is a random variable with density function satisfying the heat equation. In this paper, we consider the high dimensional case and prove that the Fisher information is square root convex in dimension two, that is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mi>d</mi><mn>2</mn></msup><mrow><mi>d</mi><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></mstyle><msqrt><msub><mi>I</mi><mi>X</mi></msub></msqrt><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The proof is based on the semidefinite programming approach.
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