Document Zbl 1245.52001

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Math.</a> <a href="/?q=in%3A186116" title="Articles in this Issue">205, No. 1, 33-83 (2006)</a>. </div> <div class="abstract">Summary: The $L_p$-Minkowski problem introduced by Lutwak is solved for $p\geq n+1$ in the smooth category. The relevant Monge-Ampère equation \[ \det(h_{ij}+h\delta_{ij})=fh^{p-1} \tag{0.1} \] is solved for all $p>1$ where $h_{ij}$ denotes convariant differentiation. The same equation for $p<1$ is also studied and solved for $p\in(-n-1,1)$. When $p=-n-1$ the equation is interpreted as a Minkowski problem in centroaffine geometry. A Kazdan-Warner-type obstruction for this problem is obtained.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5054348">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5054348">Cited in <strong>242</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A52A38" title="MSC2020">52A38</a> </td> <td class="space"> Length, area, volume and convex sets (aspects of convex geometry) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J20" title="MSC2020">35J20</a> </td> <td class="space"> Variational methods for second-order elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35J60" title="MSC2020">35J60</a> </td> <td class="space"> Nonlinear elliptic equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52A21" title="MSC2020">52A21</a> </td> <td class="space"> Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52A39" title="MSC2020">52A39</a> </td> <td class="space"> Mixed volumes and related topics in convex geometry </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52A40" title="MSC2020">52A40</a> </td> <td class="space"> Inequalities and extremum problems involving convexity in convex geometry </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A53A15" title="MSC2020">53A15</a> </td> <td class="space"> Affine differential geometry </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3A%5C%28L_p%5C%29-Minkowski+problem">$L_p$-Minkowski problem</a>; <a href="/?q=ut%3Acentroaffine+Minkowski+problem">centroaffine Minkowski problem</a>; <a href="/?q=ut%3AMonge-Amp%C3%A8re+equation">Monge-Ampère equation</a>; <a href="/?q=ut%3ABlaschke-Santalo+inequality">Blaschke-Santalo inequality</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1245.52001" data-ciurl="/ci/05054348" data-biburl="/bibtex/05054348.bib" data-amsurl="/amsrefs/05054348.bib" data-xmlurl="/xml/05054348.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05054348.pdf" title="Zbl 1245.52001 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.aim.2005.07.004" aria-label="DOI for “The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry”" title="10.1016/j.aim.2005.07.004">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ai, J.; Chou, K. S.; Wei, J., Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differential Equations, 13, 311-337 (2001) · <a href="/1086.35035" class="nowrap">Zbl 1086.35035</a></td> </tr><tr> <td>[2]</td> <td class="space">Andrews, B., Evolving convex curves, Cal. Var. Partial Differential Equations, 7, 315-371 (1998) · <a href="/0931.53030" class="nowrap">Zbl 0931.53030</a></td> </tr><tr> <td>[3]</td> <td class="space">Andrews, B., Gauss curvature flows: the fate of the rolling stones, Invent. Math., 138, 151-161 (1999) · <a href="/0936.35080" class="nowrap">Zbl 0936.35080</a></td> </tr><tr> <td>[4]</td> <td class="space">Andrews, B., Motion of hypersurfaces by Gauss curvature, Pacific J. Math., 195, 1-34 (2000) · <a href="/1028.53072" class="nowrap">Zbl 1028.53072</a></td> </tr><tr> <td>[5]</td> <td class="space">Andrews, B., Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc., 16, 443-459 (2003) · <a href="/1023.53051" class="nowrap">Zbl 1023.53051</a></td> </tr><tr> <td>[6]</td> <td class="space">Caffarelli, L. A., A localization property of viscosity solutions to the Monge-Ampire equation and their strict convexity, Ann. Math., 131, 129-134 (1990) · <a href="/0704.35045" class="nowrap">Zbl 0704.35045</a></td> </tr><tr> <td>[7]</td> <td class="space">Caffarelli, L. A., Interior $W^{2, p}$-estimate for solutions of the Monge-Ampère equations, Ann. Math., 131, 135-150 (1990) · <a href="/0704.35044" class="nowrap">Zbl 0704.35044</a></td> </tr><tr> <td>[8]</td> <td class="space">K.S. Chou, X.J. Wang, Existence of solutions to a generalized Minkowski problem, Manuscript, 1995.; K.S. Chou, X.J. Wang, Existence of solutions to a generalized Minkowski problem, Manuscript, 1995.</td> </tr><tr> <td>[9]</td> <td class="space">Chou, K. S.; Wang, X. J., The logarithmic Gauss curvature flow, Ann. Inst. H. Poincarè Anal. Non Linèaire, 17, 733-751 (2000) · <a href="/1071.53534" class="nowrap">Zbl 1071.53534</a></td> </tr><tr> <td>[10]</td> <td class="space">Chou, K. S.; Wang, X. J., A variational theory of the Hessian equation, Comm. Pure Appl. Math., 54, 1029-1064 (2001) · <a href="/1035.35037" class="nowrap">Zbl 1035.35037</a></td> </tr><tr> <td>[11]</td> <td class="space">Cheng, S. Y.; Yau, S. T., On the regularity of the $n$-dimensional Minkowski problem, Comm. Pure Appl. Math., 20, 41-68 (1977)</td> </tr><tr> <td>[12]</td> <td class="space">Firey, W. J., $p$-means of convex bodies, Math. Scand., 10, 17-24 (1968) · <a href="/0188.27303" class="nowrap">Zbl 0188.27303</a></td> </tr><tr> <td>[13]</td> <td class="space">P. Guan, C.-S. Lin, On equation $\det( u_{\mathit{ij}} + \delta_{\mathit{ij}} u) = u^p f \text{on} S^n$; P. Guan, C.-S. Lin, On equation $\det( u_{\mathit{ij}} + \delta_{\mathit{ij}} u) = u^p f \text{on} S^n$</td> </tr><tr> <td>[14]</td> <td class="space">C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contribution to Analysis, Academic Press, New York, 1974, pp. 245-272.; C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contribution to Analysis, Academic Press, New York, 1974, pp. 245-272. · <a href="/0298.35018" class="nowrap">Zbl 0298.35018</a></td> </tr><tr> <td>[15]</td> <td class="space">Lutwak, E., The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem, J. Differential Geom., 38, 131-150 (1993), II: affine and geominimal surface areas, Adv. Math. 118 (1996) 244-294 · <a href="/0788.52007" class="nowrap">Zbl 0788.52007</a></td> </tr><tr> <td>[16]</td> <td class="space">Lutwak, E.; Oliker, V., On the regularity of solution to a generalization of the Minkowski problem, J. Differential Geom., 40, 227-246 (1995) · <a href="/0867.52003" class="nowrap">Zbl 0867.52003</a></td> </tr><tr> <td>[17]</td> <td class="space">Lutwak, E.; Yang, D.; Zhang, G., On the $L_p$-Minkowski problem, Trans. Amer. Math. Soc., 356, 4359-4370 (2004) · <a href="/1069.52010" class="nowrap">Zbl 1069.52010</a></td> </tr><tr> <td>[18]</td> <td class="space">Lutwak, E.; Yang, D.; Zhang, G., Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom., 62, 17-38 (2002) · <a href="/1073.46027" class="nowrap">Zbl 1073.46027</a></td> </tr><tr> <td>[19]</td> <td class="space">K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, New York, 1994.; K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, New York, 1994. · <a href="/0834.53002" class="nowrap">Zbl 0834.53002</a></td> </tr><tr> <td>[20]</td> <td class="space">Pogorelov, A. V., The Multidimensional Minkowski Problem (1978), Wiley: Wiley New York · <a href="/0387.53023" class="nowrap">Zbl 0387.53023</a></td> </tr><tr> <td>[21]</td> <td class="space">R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993.; R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. · <a href="/0798.52001" class="nowrap">Zbl 0798.52001</a></td> </tr><tr> <td>[22]</td> <td class="space">Tzitzéica, G., Sur une nouvelle classe de surfaces, Rend. Circ. Mat. Palermo, 25, 180-187 (1908), 28 (1909) 210-216 · <a href="/39.0685.05" class="nowrap">JFM 39.0685.05</a></td> </tr><tr> <td>[23]</td> <td class="space">Urbas, J., Complete noncompact self-similar solutions of Gauss curvature flows. I. Positive powers, Math. Ann., 311, 251-274 (1998) · <a href="/0910.53043" class="nowrap">Zbl 0910.53043</a></td> </tr><tr> <td>[24]</td> <td class="space">Urbas, J., Complete noncompact self-similar solutions of Gauss curvature flows. II. Negative powers, Adv. Differential Equations, 4, 323-346 (1999) · <a href="/0957.53033" class="nowrap">Zbl 0957.53033</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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