View source for Brunn-Minkowski theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177202.png" /> be convex sets in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177203.png" />-dimensional Euclidean space; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177205.png" />, be the set of points which divide segments with end points at any points of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177207.png" /> in the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177208.png" /> (a linear combination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b0177209.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772010.png" />); and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772011.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772012.png" />-th power root of the volume of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772013.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772014.png" /> is a concave function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772015.png" />, i.e. the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772016.png" /></td> </tr></table>

is valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772017.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772018.png" /> is linear (and the inequality then becomes an equality) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017720/b01772020.png" /> are homothetic. The Brunn–Minkowski theorem can be generalized to linear combinations of several convex sets. It is used to solve extremal and uniqueness problems. It was discovered by H. Brunn in 1887, and completed and rendered more precise in 1897 by H. Minkowski.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Busemann,   "Convex surfaces" , Interscience  (1958)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Hadwiger,   "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer  (1957)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Leichtweiss,   "Konvexe Mengen" , Springer  (1979)</TD></TR></table>