We study a curvature flow on smooth, closed, strictly convex hypersurfaces in |$\mathbb{R}^n$|⁠, which commutes with the action of |$SL(n)$|⁠. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in |$\mathbb{R}^n$| with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

中文翻译：

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从一个凸体到另一个凸体指定中心仿射曲率

我们在| $ \ mathbb {R} ^ n $ |⁠中研究光滑，封闭且严格凸的超曲面上的曲率流，该曲面与| $ SL（n）$ |⁠的作用相对应。该流将初始超曲面缩小到一个点，如果重新缩放以包围恒定体积的域，则|| \\ mathbb {R} ^ n $ |中将是一个平滑，闭合，严格凸的超曲面。中心仿射曲率与固定超曲面的中心仿射曲率成比例，但并不总是相等。我们概述了该结果对凸体的几何形状和对数Minkowski不等式的一些后果。