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Logarithmically concave measure
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<h2 id="examples">Examples</h2> <p>The <a href="/facts/Brunn%E2%80%93Minkowski_theorem/gIvhLupE">Brunn–Minkowski inequality</a> asserts that the <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a> is log-concave. The restriction of the Lebesgue measure to any <a href="/facts/Convex_set/vdAuJRJl">convex set</a> is also log-concave. </p><p>By a theorem of Borell,<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> a probability measure on R^d is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a <a href="/facts/Logarithmically_concave_function/KWp07q9b">logarithmically concave function</a>. Thus, any <a href="/facts/Gaussian_measure/WRL0Sdx5">Gaussian measure</a> is log-concave. </p><p>The <a href="/facts/Pr%C3%A9kopa%E2%80%93Leindler_inequality/QVpojlpk">Prékopa–Leindler inequality</a> shows that a <a href="/facts/Convolution/PVPrdz9J">convolution</a> of log-concave measures is log-concave. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Convex_measure/CYXLJyIQ">Convex measure</a>, a generalisation of this concept</li> <li><a href="/facts/Logarithmically_concave_function/KWp07q9b">Logarithmically concave function</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Prékopa, A. (1980). "Logarithmic concave measures and related topics". Stochastic programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974). London-New York: Academic Press. pp. 63–82. MR 0592596. <a href="/wiki/Andr%C3%A1s_Pr%C3%A9kopa" target="_blank">/wiki/Andr%C3%A1s_Pr%C3%A9kopa</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Borell, C. (1975). "Convex set functions in d-space". Period. Math. Hungar. 6 (2): 111–136. doi:10.1007/BF02018814. MR 0404559. S2CID 122121141. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>