Geodesic
On the Order of the Lebesgue Constants for Interpolation by Algebraic Polynomials from Values at Uniform Nodes of a Simplex
Matematičeskie zametki, Tome 77 (2005) no. 6, pp. 814-831.

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For interpolation processes by algebraic polynomials of degree $n$ from values at uniform nodes of an $m$-simplex, where $m\ge2$, we obtain the order of growth in $n$ of the Lebesgue constants, which coincides with that in the one-dimensional case for which Turetskii obtained an asymptotics earlier. 
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N. V. Baidakova. On the Order of the Lebesgue Constants for Interpolation by Algebraic Polynomials from Values at Uniform Nodes of a Simplex. Matematičeskie zametki, Tome 77 (2005) no. 6, pp. 814-831. http://geodesic.mathdoc.fr/item/MZM_2005_77_6_a1/

[1] Syarle F., Metod konechnykh elementov dlya ellipticheskikh zadach, Mir, M., 1980 | MR

[2] Kurosh A. G., Kurs vysshei algebry, Nauka, M., 1971

[3] Nicolaidis R. A., “On the class of finite elements generated by Lagrange interpolation”, SIAM J. Numer. Anal., 9:3 (1972), 435–445 | DOI | MR

[4] Turetskii A. Kh., Teoriya interpolirovaniya v zadachakh, Vysheishaya shkola, Minsk, 1968