References
T. Bonnesen andW. Fenchel, Theorie der konvexen Körper,Ergebnisse der Mathematik, 3, Heft 1 (1934).
C. Carathéodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen,Rendiconti di Palermo, 32 (1911), 193–217.
C. Carathéodory,Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Leipzig, 1935.
F. Gantmakher andM. Krein,Oscillatory matrices and kernels and small vibrations of mechanical systems (in Russian), 2nd Ed., Moscow, 1950.
W. Gustin, Sets of finite planar order,Duke Math. Journal, 14 (1947), 51–66.
J. Hadamard, Résolution d'une question relative aux déterminants,Bulletin des sciences mathématiques, (2), 17 (1893), 240–248.
J. Hjelmslev, Introduction à la théorie des suites monotones,Oversigt over det Kgl. Danske Videnskabernes Selskabs Forhandlinger, 1914, 1–74.
A. Hurwitz, Sur le problème des isopérimétres,Comptes Rendus, 132 (1901), 401–403; also inMath. Werke 1, 490–491.
A. Marchaud, Sur les continus d'ordre borné,Acta Mathematica 55 (1930), 67–115.
J. von Neumann andI. J. Schoenberg, Fourier integrals and metric geometry,Transactions of the Amer. Math. Society, 50 (1941), 226–251.
G. Pólya andG. Szegö,Aufgaben und Lehrsätze aus der Analysis, II, Berlin, 1925.
G. Pólya On the mean-value theorem corresponding to a given linear homogeneous differential equation,Transactions of the Amer. Math. Society, 24 (1922), 312–324.
Peter Scherk, Über differenzierbare Kurven und Bögen,Časopis pro pěstováni Matematiky a Fysiky 66 (1937), 165–191.
I. J. Schoenberg andAnne Whitney, A theorem on polygons inn dimensions with applications to variation-diminishing and cyclic variation-diminishing linear transformations,Compositio Mathematica, 9 (1951), 141–160.
I. J. Schoenberg, On smoothing operations and their generating functions,Bulletin of the Amer. Math. Society, 59 (1953), 199–230.
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This work was performed on a National Bureau of Standards contract with the University of California, Los Angeles, and was sponsored (in part) by the Office of Scientific Research, USAF.
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schoenberg, I.J. An isoperimetric inequality for closed curves convex in even-dimensional euclidean spaces. Acta Math. 91, 143–164 (1954). https://doi.org/10.1007/BF02393429
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DOI: https://doi.org/10.1007/BF02393429